Classes of Operator-Smooth Functions II. Operator ... - Springer Link

Integr. equ. oper. theory 49 (2004), 165–210 0378-620X/020165-46, DOI 10.1007/s00020-002-1201-0 c 2004 Birkh¨ auser Verlag Basel/Switzerland

Integral Equations and Operator Theory

Classes of Operator-Smooth Functions II. Operator-Differentiable Functions E. Kissin and V.S. Shulman Abstract. This paper studies the spaces of Gateaux and Frechet Operator Differentiable functions of a real variable and their link with the space of Operator Lipschitz functions. Apart from the standard operator norm on B(H), we consider a rich variety of spaces of Operator Differentiable and Operator Lipschitz functions with respect to symmetric operator norms. Our approach is aimed at the investigation of the interrelation and hierarchy of these spaces and of the intrinsic properties of Operator Differentiable functions. We apply the obtained results to the study of the functions acting on the domains of closed *-derivations of C*-algebras and prove that Operator Differentiable functions act on all such domains. We also obtain the following modification of this result: any continuously differentiable, Operator Lipschitz function acts on the domains of all weakly closed *-derivations of C*-algebras. Mathematics Subject Classification (2000). Primary 47A56, 46L57; Secondary 47L20. Keywords. Operator, Lipschitz, differentiable, functions, derivations, C*-algebras.

Introduction This paper studies the spaces of Gateaux and Frechet Operator Differentiable functions of a real variable. It continues the study of Operator-smooth functions started in [19], where Operator Lipschitz functions were investigated. Let B(H) be the algebra of all bounded operators on a Hilbert space H. Recall that any bounded, Borel function g on a set α ⊆ R defines, via the Spectral Theorem, a map g˜ : T → g(T ) from the set of all selfadjoint operators with spectrum in α into B(H). Various smoothness conditions when imposed on these maps characterize interesting and important classes of functions. In particular, the condition that g˜ is Gateaux (resp. Frechet) differentiable defines the class of The first author is grateful to the Leverhulme Trust for the award of a Research Fellowship.

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Gateaux (resp. Frechet) Operator Differentiable functions; the condition that g˜ is Lipschitzian defines the class of Operator Lipschitz functions. If, apart from the standard operator norm on B(H), one also considers other unitarily invariant operator norms - symmetric operator norms (see Section 1) and the spaces of Gateaux (Frechet) Operator Differentiable functions and Operator Lipschitz functions with respect to these norms, one obtains a rich variety of classes of functions. Thus the Operator Theory suggests its own scale of smoothness of functions and defines naturally new functional spaces. Much work has been done to relate the “operator” smoothness of functions to the traditional “scalar” smoothness conditions. Following on from the paper of Daletskii and Krein [8], there have been numerous articles by Birman and Solomyak [3, 4], Davies [9], Farforovskaya [10, 11] and others in which the authors determine the smoothness of Operator Differentiable and Operator Lipschitz functions. This work culminated in the result of Peller [23] which placed the class of Gateaux 1 (a, b) Operator Differentiable functions on [a, b] between two Besov spaces B∞1 1 and B11 (a, b). Later Arazy, Barton and Friedman [2] constructed another func1 tional space, wider than B∞1 (a, b), contained in the class all Gateaux Operator Differentiable functions. Another type of “operator” smoothness conditions on a function g arises if we assume that g˜ preserves some class of Differential Algebras - operator analogues of algebras of differentiable functions - introduced by Blackadar and Cuntz [5] (see also [16]) and, in particular, the domains of all unbounded *-derivations of C ∗ -algebras. In this setting the interplay of the “operator” smoothness with the “scalar” smoothness was investigated by Powers [24], McIntosh [21], Bratteli, Elliott and Jorgensen [6] and others who studied the functions preserving the domains of unbounded *-derivations of C ∗ -algebras. Significant similarities between various properties of these functions, of Gateaux Operator Differentiable and of Operator Lipschitz functions point to a close link between these classes. The first step in clarifying this link was made by Arazy, Barton and Friedman [2] who proved that Gateaux Operator Differentiable functions act on the domains of the generators of one-parameter groups of automorphisms of C ∗ -algebras. One of the motivations behind our work was the desire to find out whether this result can be extended to all closed *-derivations of C ∗ -algebras. Theorem 8.1 of the present paper provides this extension. Our work was also stimulated by the following intriguing problem: do the spaces of all Operator Differentiable and all Operator Lipschitz functions coincide? It has transpired that not only are there Operator Lipschitz functions which are not Operator Differentiable but that Operator Differentiable functions form a closed separable subspace in the non-separable space of all Operator Lipschitz functions (see Theorem 7.12). As in [19], our approach is aimed at the investigation of the intrinsic properties of Operator Differentiable functions and at the interrelation and hierarchy of various spaces of “operator” smooth functions. We do not consider their relation to the classical spaces of smooth functions as was done in the papers of Birman and

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Solomyak, Davies, Farforovskaya, Peller, Arazy, Barton and Friedman and others cited above. Consequently, we do not use the elaborate technique of the Function Theory and the Double Operator Integral machinery in spite of the fact that they have been heavily used in the previous works since the publication by Daletskii and Krein of their paper, where the Gateaux derivative was expressed as a Double Operator Integral. We proceed now with a brief description of the results of the paper. In Section 2 we introduce the notions of Gateaux and Frechet J-Differentiable functions (by J we denote symmetrically normed ideals of B(H)) and consider some well-known sufficient conditions for functions to belong to these classes. The main result of Section 3 is Theorem 3.6. It establishes that, for a separable ideal J, the space of Gateaux J-Differentiable functions consists of all J-Lipschitz functions, differentiable in the usual “scalar” sense. For J = B(H), however, this is not true (see Proposition 4.4) and we show in Theorem 3.7 that Operator Lipschitz functions (Johnson and Williams [15] proved that they are always differentiable in the usual sense) are exactly those functions which are Gateaux Differentiable along compact directions. In Section 4 we prove that a function is Frechet J-Differentiable if it is Frechet J-Differentiable along “finite rank” directions. More importantly, we also show that, for J = B(H), the notions of the Gateaux and Frechet J-Differentiabilities of functions coincide; we call such functions Operator Differentiable. Section 5 contains the proof of the fact that the “uniform continuity on compacts” of the Gateaux derivative characterizes Frechet J-Differentiable functions in the class of Gateaux J-Differentiable functions. Making use of this, we establish in Section 6 a formula for the Gateaux derivative of the convolution of two functions - a useful technical tool for the subsequent work. In Section 5 we also consider Gateaux and Frechet Operator Differentiable functions on finite-dimensional spaces H. Since all ideals in this case coincide and all norms are equivalent, there is only one space of Operator Lipschitz functions, one space of Gateaux and one space of Frechet Operator Differential functions. It was noted in [19] that if dim H < ∞ then a function is Operator Lipschitzian if and only if it is Lipschitzian in the usual sense. In Section 5 we show that if dim H < ∞ then, like in the infinite dimensional case, the spaces of Gateaux and Frechet Operator Differentiable functions coincide. Moreover, these functions can be characterized in the standard “scalar” terms: a function is Operator Differentiable if and only if it is differentiable in the usual sense; its Gateaux derivative is continuous if and only if its usual derivative is continuous. This strengthens the well-known result (see, for example [13]) that continuously differentiable functions are Frechet Operator differentiable (if H is finite-dimensional). In Section 7 we study topological properties of the spaces of J-Lipschitz functions and of Gateaux and of Frechet J-Differentiable functions. We show that Gateaux J-Differentiable functions constitute a closed subspace in the space of all J-Lipschitz functions and that both spaces are non-separable. The subspace of

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all Frechet J-Differentiable functions is also closed in the space of all J-Lipschitz functions and we prove that, for J = B(H), this subspace is separable. In the concluding Section 8 we apply the results of the preceding sections to the study of the functions acting on the domains of closed *-derivations of C ∗ algebras. As mentioned above, we extend the result of Arazy, Barton and Friedman [2] and prove that Operator Differentiable functions act on all such domains (Theorem 8.1). In Theorem 8.4 we show that the space of functions acting on the domains of all weakly closed *-derivations of C ∗ -algebras coincides with the space of all Operator Lipschitz functions. The latter result matches well with the view that the domains of weakly closed *-derivations of C ∗ -algebras are non-commutative analogues of Lipschitz algebras (see [7] and [28]). When this paper was ready for submission, our attention was drawn to the paper of G.K. Pedersen [22] which, for the case J = B(H), contains the proofs of our Proposition 6.1, Theorems 5.2 and 7.9 and a sketch of the proof of Theorem 8.1. It is worth noting, however, that most of our results for this case were already obtained in 1995 (see the short communication [17]). The authors are grateful to C.J.K. Batty, V.I. Burenkov, E.B. Davies, Yu.B. Farforovskaya, V.I. Ovchinnikov and V.V. Peller for helpful discussions. They would also like to thank the referee for his perceptive, helpful comments and especially for his questions about the finite-dimensional case. The authors are also grateful to T.V. Shulman for drawing their attention to a weaker version of Lemma 5.5, proved in her work [27].

1. Preliminaries Let (B(H), ·) be the algebra of all bounded operators on a Hilbert space H, let C(H) be the ideal of all compact operators and F be the ideal of all finite rank operators. A two-sided ideal J of B(H) is symmetrically normed (we write s. n. ideal) if it is a Banach space with respect to a norm |·|J , if |AXB|J ≤ A|X|J B, for A, B ∈ B(H) and X ∈ J, and if |X|J = X for rank one operators. By Calkin’s theorem, F ⊆ J ⊆ C(H). If X ∈ J and U is an isometry (U ∗ U = 1) then X ≤ |X|J = |X ∗ |J and |U X|J = |XU ∗ |J = |X|J , for X ∈ J. By J˜ we denote the unitization J˜ = (C1) + J of J with the norm |λ1 + A|J˜ = |λ| + |A|J , for λ ∈ C and A ∈ J. Let c0 be the space of all sequences of real numbers converging to 0, cˆ be the subspace of c0 which consists of all sequences with finite number of nonzero elements and let Φ be the set of symmetric norming functions on cˆ (see [14, §III.3]). For ξ = (ξi ) ∈ c0 , let ξ (n) = (ξ1 , . . . , ξn , 0, . . .). For φ ∈ Φ, the sequence φ(ξ (n) ) does not decrease. Set φ(ξ) = lim φ(ξ (n) ) and cφ = {ξ ∈ c0 : lim φ(ξ) < ∞}.

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For A ∈ C(H), let s(A) = (si (A)) be the non-increasing sequence of all eigenvalues of (A∗ A)1/2 (multiple eigenvalues are repeated according to multiplicity). For φ ∈ Φ, the set J = J φ = {A ∈ C(H) : s(A) ∈ cφ } with norm |A|J = φ(s(A)) is a s. n. ideal (see [14]). The closure J0φ of F in norm |·|J is a separable s. n. ideal and J0φ ⊆ J φ . It follows from Theorem III.6.2 of [14] that a s. n. ideal is separable if and only if it coincides with some J0φ . For φ ∈ Φ, one can define the adjoint s. n. function φ∗ (see [14, §III.11]). The s. n. ideal J φ∗ is isometrically isomorphic to the dual space of the separable ideal J0φ . For many s. n. functions φ, the ideals J φ and J0φ coincide. An important class of such functions consists of the s. n. functions ∞ φp (ξ) = ( |ξi |p )1/p for 1 ≤ p < ∞, and φ∞ (ξ) = sup |ξi |. i=1

The corresponding ideals are called Schatten classes. We denote them by S p and their norms by |·|p . In particular, S ∞ = C(H) and |·|∞ = ·. We usually denote B(H) by S. Then (see [14]) S1 ⊆ Sp ⊆ Sq ⊆ S∞ ⊂ S and |A|p ≥ |A|q ≥ A, if p ≤ q and A ∈ S p . If H is finite-dimensional then all s. n. ideals coincide with B(H), but their norms are different. Remark 1.1. In this paper we only consider separable Hilbert spaces. Let H be an infinite dimensional space and let J be a s. n. ideal of B(H). 1) For any unitary U on H, we have U J = JU = J. This allows us to identify s. n. ideals acting on different infinite dimensional Hilbert spaces: if K is another such space and V is an isometry from H onto K, then the ideal V JV ∗ of B(K) with the norm |V AV ∗ |V JV ∗ = |A|J , for A ∈ J, is isometrically isomorphic to J, does not depend on the choice of V and, hence, can be also denoted by the same symbol J (or J(K) if it is necessary to underline the space). Since the sequences s(A) and s(V AV ∗ ) coincide for all A ∈ J, the fact that J is J φ or J0φ does not depend on the underlying Hilbert space. 2) Let K = H n be the orthogonal sum of n copies of H. Any operator in the ideal J(K) can be represented as a block-matrix with all entries from J. If n < ∞, then J(K) consists of all block-matrices with entries from J. If A = (Aij ) ∈ J(K) is such that all entries, apart from one entry Akm , are zero operators, then |A|J(K) = |Akm |J . Notation. 1) For any subset σ of R, we denote by Ssa (σ) the set of all selfadjoint operators with spectrum in σ, by Jsa (σ) the set of all selfadjoint operators in J with spectrum in σ and by Fsa (σ) the set of all selfadjoint finite rank operators with spectrum in σ.

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2) We denote by Γ an open subset and by α a compact subset of R. It is easy to check that the set Jsa (α) is closed and the set Jsa (Γ) is open.

2. Gateaux and Frechet J-differentiable functions We start with the definition of Gateaux and Frechet differentiable maps suitable for our purposes. Definition 2.1. Let J be a s. n. ideal including S, let Z be a linear manifold in Jsa and let F be a map from Ssa (Γ) to S. (i) The map F is J-Differentiable at A ∈ Ssa (Γ) along Z ∈ Jsa , if there is Zˆ ∈ J such that, for any > 0, there exists β > 0 such that |t| ≤ β implies ˆ J ≤ |t|. F (A + tZ) − F (A) ∈ J and |F (A + tZ) − F (A) − tZ|

(2.1)

(ii) The map F is Gateaux J-Differentiable at A ∈ Ssa (Γ) along Z, if it is J-Differentiable at A along all Z in Z and the map FA∇ : Z → Zˆ is a bounded linear operator from Z to J. If, in addition, for any > 0, there exists β > 0 such that, F (A + Z) − F (A) ∈ J and |F (A + Z) − F (A) − FA∇ (Z)|J ≤ |Z|J , for all Z ∈ Z with |Z|J ≤ β, then F is Frechet J-Differentiable at A along Z. The operator FA∇ is called the G-derivative of F at A along Z. Clearly, any Frechet J-Differentiable map is also Gateaux J-Differentiable. If ˜ J˜ is the unitization of J, we can similarly define Gateaux (Frechet) J-Differentiability of a map at A along a manifold Z ⊆ J˜sa . By the Spectral Theorem, any continuous function g on Γ defines a map A → g(A), also denoted by g, from Ssa (Γ) into S. We say that g is Gateaux (Frechet) J-Differentiable at A ∈ Ssa (Γ) along a linear manifold Z ⊆ Jsa if the corresponding map is Gateaux (Frechet) J-Differentiable at A along Z. If a function g is Gateaux (Frechet) J-Differentiable (i) at A along Jsa , we say that g is Gateaux (Frechet) J-Differentiable at A; (ii) at all A ∈ Ssa (Γ) along Jsa , we say that g is Gateaux (Frechet) JDifferentiable on Γ. The next technical lemma illustrates the notions and supplies us with many examples of Gateaux and Frechet J-Differentiable functions. Lemma 2.2. Let g be a continuous function on R, let g ∈ L1 (R) and let gˆ(s) be its Fourier transform. (i) g is Gateaux J-Differentiable on R for any s. n. ideal or S, if ∞ |sˆ g (s)|ds < ∞ (2.2) −∞

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(in particular, if g ∈ L2 (R) (see the proof of Corollary 2.3)), and 1 ∞ 1 ∇ (X) = (−is)ˆ g (s)( e−isuA Xe−is(1−u)A du)ds gA 2π −∞ 0 for X ∈ Jsa . (ii) g is Frechet J-Differentiable on R, for any s. n. ideal or S, if ∞ |s2 gˆ(s)|ds < ∞,

171

(2.3)

(2.4)

−∞

in particular, if g ∈ L2 (R) (see the proof of Corollary 2.3)). Proof. Denote by T (X) the right-hand side of (2.3). Then (2.2) implies that T is a bounded operator on J with ∞ 1 T J ≤ |sˆ g (s)|ds. (2.5) 2π −∞ For any A ∈ Ssa (R),

∞ 1 g(A) = eisA gˆ(s)ds. 2π −∞ For A, B ∈ B(H), we have (see, for example [9]) 1 A B euA (A − B)e(1−u)B du. e −e =

(2.6)

0

Making use of (2.6), we obtain from (2.3) 1 | (g(A + tX) − g(A)) − T (X)|J t 1 ∞ 1 ≤ |sˆ g (s)| |(e−isu(A+tX) − e−isuA )Xe−is(1−u)A |J duds. 2π −∞ 0

(2.7)

Since, for any s and u, e−isu(A+tX) − e−isuA → 0, as t → 0 , we obtain from (2.2) and the Dominated Convergence Theorem that | 1t (g(A + tX) − g(A)) − T (X)|J → 0, as t → 0. This proves (i). To prove (ii), set t = 1 in (2.7). Applying (2.6) and (2.4), we have ∞ 1 |s|2 2 |X|J ds = o(|X|J ). |ˆ g (s)| |g(A + X) − g(A) − T (X)|J ≤ 2π 2 −∞ ∇ Thus g is Frechet J-Differentiable at A along Jsa and T = gA .

While verifying Frechet or Gateaux J-Differentiability of a function at A, one only deals with the restriction of the function to a neighbourhood of Sp(A). Therefore the conditions of Lemma 2.2 may be understood in the local sense: a function is Gateaux (resp. Frechet)J-Differentiable if on any finite interval it coincides with a function satisfying (2.2) (resp. (2.4)). The conditions of Lemma 2.2, far from being necessary, can be considerably improved. Condition (2.2), for example, guarantees, in fact, Frechet J-Differentiability of the function (see Introduction where much stronger results of Birman

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and Solomyak, Davies, Peller and others are discussed). Moreover, as we will see in Theorem 4.3, for the most important case J = S, all Gateaux Differentiable functions are also Frechet Differentiable. However, for our further use the results of Lemma 2.2 are sufficient. Consider the following algebra norm on the commutative algebra C (2) (α): 1 g2,α = gα + g α + g α , (2.8) 2 where gα = sup |g(t)|. t∈α

∇ J the norm of the G-deriIf g is Gateaux J-differentiable at A, we denote by gA ∇ vative. We deduce now an estimate for gA J in terms of the norm g2,α .

Corollary 2.3. For any compacts α, β ⊂ Γ such that α ⊂ int(β), there exists a constant C(α, β) > 0 such that ∇ J ≤ C(α, β)g2,β , gA

for each g ∈ C

(2)

(2.9)

(Γ) and each A ∈ Ssa (α), where J is any s. n. ideal or S.

Proof. Fix a smooth function ϕ such that supp(ϕ) ⊆ β and that ϕ(t) ≡ 1 in a neighborhood U of α. Let µ be the measure of β. Set C(α, β) = 4µ1/2 ϕ2,β . If supp(g) ⊆ β then (see, for example, [26, Theorem 3.3.7]) ∞ 1 ∞ ∞ 2 1/2 2 1/2 |sˆ g (s)|ds ≤ ( s ds) ( |ˆ g (s)| ds) + 2( |s2 gˆ(s)|2 ds)1/2 . −∞

−1

−∞

−∞

Since −s2 gˆ(s) is the Fourier transform of g , ∞ ∞ 2 1/2 |sˆ g (s)|ds ≤ 2π( |g(t)| dt) + 4π( −∞

−∞

∞

−∞

|g (t)|2 dt)1/2

≤ 2π sup |g(t)|µ1/2 + 4π sup |g (t)|µ1/2 ≤ 8πµ1/2 g2,β . t∈β

t∈β

∇ Hence it follows from (2.3) and (2.5) that gA J ≤ C(α, β) g2,β . Let supp(g) do not lie in β. Since g(A + X) = g(A + X)ϕ(A + X), whenever ∇ Sp(A + X) ⊂ U , we have that gA = (gϕ)∇ A . Since supp(gϕ) ⊆ β, we have ∇ 1/2 J = (gϕ)∇ gϕ2,β ≤ 4µ1/2 ϕ2,β g2,β = C(α, β)g2,β . gA A J ≤ 4µ

We see that sufficiently smooth functions are Gateaux J-Differentiable and Frechet J-Differentiable. We will show that in many cases (but not always - see the discussion before Lemma 3.4) the usual differentiability is necessary for a function to be J-Differentiable. Proposition 2.4. Let g be a continuous function on Γ and let J be an ideal or S. (i) Let λ be an eigenvalue of A ∈ Ssa (Γ) and let Q be the projection on the corresponding eigenspace. If g is J-Differentiable at A along Q, then g (λ) exists and ∇ (Q) = g (λ)Q. (2.10) gA

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(ii) If g is Frechet J-Differentiable at every A ∈ J˜sa (Γ) along Fsa , then g is continuously differentiable on Γ. (iii) If g is S-Differentiable at every A ∈ Ssa (Γ) along 1H , then g is continuously differentiable on Γ. Proof. Part (i) follows from the equality g(A + tQ) − g(A) = (g(λ + t) − g(λ))Q.

(2.11)

By (i), g is differentiable on Γ. If g is not continuous at x ∈ Γ, there exist C > 0 and (an )∞ n=2 in Γ converging to x such that |g (an ) − g (x)| > C. Choose an ) in H and let P be the projections onto Cen . Set a1 = x orthonormal basis (e n n a P . Then Sp(A) ⊂ Γ. Without loss of generality we may assume and A = ∞ n=1 n n ˜ that an converge to x so fast that A − x1 ∈ Jsa , that is, A ∈ Jsa . For < C/2, let δ > 0 be such that, for X ∈ Fsa with |X|J < δ, ∇ (X)|J < |X|J . |g(A + X) − g(A) − gA

Choose n > 1 such that |x − an | < δ and set Z = (an − x)P1 , Y = (x − an )Pn . Then |Z|J = |Y |J = |x − an | < δ. Therefore ∇ (Z)|J < |x − an |, |g(A + Z) − g(A) − gA ∇ |g(A + Y ) − g(A) − gA (Y )|J < |x − an |.

Since, by (2.10), ∇ ∇ gA (Z) = (an − x)gA (P1 ) = (an − x)g (a1 )P1 = (an − x)g (x)P1 , ∇ ∇ gA (Y ) = (x − an )gA (Pn ) = (x − an )g (an )Pn ,

it follows from (2.11) and from the above inequalities that |[g(x + (an − x)) − g(x)]P1 − (an − x)g (x)P1 |J < |x − an |, |[g(an + (x − an )) − g(an )]Pn − (x − an )g (an )Pn |J < |x − an |. Therefore |g(an ) − g(x) − (an − x)g (x)| < |x − an |, |g(x) − g(an ) − (x − an )g (an )| < |x − an |, so that |g (x) − g (an )| < 2 < C. This contradiction proves part (ii). Let x, an , Pn and A be the same as in (ii). Since g is S-Differentiable at A along 1H then, for any > 0, there is δ > 0 such that ∇ g(A + t1H ) − g(A) − tgA (1H ) ≤ |t|, for |t| ≤ δ.

(2.12)

Hence ∇ Pn (g(A + t1H ) − g(A) − tgA (1H ))Pk ≤ |t|. ∇ ∇ (1H )Pk ≤ |t| which implies Pn gA (1H )Pk = 0. For For n = k, we have tPn gA n = k, we obtain that ∇ (g(an + t) − g(an ))Pn − tPn gA (1H )Pn ≤ |t|.

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∇ This implies Pn gA (1H )Pn = g (an )Pn . Therefore we obtain from (2.12) that

|g(an + t) − g(an ) − tg (an )| ≤ |t|,

(2.13)

for all n and |t| ≤ δ. Let |a1 − ak | ≤ δ, for k ≥ N . Replacing t by a1 − ak in (2.13) and setting n = k, we have |g(a1 ) − g(ak ) − (a1 − ak )g (ak )| ≤ |a1 − ak |.

(2.14)

Replacing t by ak − a1 in (2.13) and setting n = 1, we obtain |g(ak ) − g(a1 ) − (ak − a1 )g (a1 )| ≤ |ak − a1 |. Combining this with (2.14) yields |g (ak ) − g (a1 )| ≤ 2, for k ≥ N. Hence g (ak ) converge to g (x). Since this holds for any sequence converging to x, g is continuous at x. By Proposition 2.4, any Gateaux J-Differentiable function on Jsa (Γ) along Fsa is differentiable in the usual sense. The fact that Frechet S-Differentiable functions are continuously differentiable was proved by Widom [29]. In Section 5 we show that such functions also have continuous G-derivative.

3. J-Lipschitz and Gateaux J-Differentiable functions In this section we prove that, for any separable ideal J = S 1 , a function is Gateaux J-Differentiable on Γ if and only if it is differentiable in the usual sense and J-Lipschitzian on Γ. To do this we will draw on the various results obtained in [19] where J-Lipschitz functions were studied. Definition 3.1. Let J be a s. n. ideal or S. (i) A continuous function g on α is J-Lipschitzian, if there exists D > 0 such that |g(A) − g(B)|J ≤ D|A − B|J for A, B ∈ Jsa (α). (3.1) If J = B(H), then 0 ∈ Sp(A), for each A ∈ Jsa , so we assume that α contains 0. (ii) A continuous function g on an open set Γ is J-Lipschitzian, if it is J-Lipschitzian on any compact subset α of Γ (the constant D in (3.1) depends on α). Using rank one projections, it is easy to see that J-Lipschitz functions are Lipschitzian in the usual sense. Furthermore it follows from Theorems 7.3 and 7.5 of [19] that if g is J-Lipschitzian, if A, B ∈ Ssa (α) and A − B ∈ J, then g(A) − g(B) ∈ J and |g(A) − g(B)|J ≤ D|A − B|J .

(3.2)

Let [A, B] = AB − BA be the commutator of A and B. The proof of the following folklore result can be found in [9] and [19]: a function g is J-Lipschitzian on α, if and only if there exists D > 0 such that |[g(A), X]|J ≤ D|[A, X]|J for A ∈ Jsa (α) and X ∈ B(H).

(3.3)

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Moreover, if J is separable or J = S, then g is J-Lipschitzian if and only if (3.1) (or (3.3)) holds for all A, B ∈ Fsa (α) and all X ∈ F. Let g be a continuous function on Γ and A ∈ Ssa (Γ). There is > 0 such that A + i[A, Y ] ∈ Ssa (Γ), for all Y ∈ Ssa ((−, )). Hence the map GA (Y ) = g(A + i[A, Y ]) is defined for all Y ∈ Ssa ((−, )). Proposition 3.2. Let g be a J-Lipschitz function on Γ, where J is J0φ , or J φ , or S. For A ∈ Ssa (Γ), the map GA is Frechet J-Differentiable at 0 along Jsa and (GA )∇ 0 (Y ) = iδg(A) (Y ) = i[g(A), Y ], for Y ∈ Jsa . Proof. Clearly, (GA )∇ 0 is a linear bounded operator on J. If Y ∈ Jsa , then i[A, Y ] ∈ Jsa . By (3.2), GA (Y ) − GA (0) = g(A + i[A, Y ]) − g(A) ∈ J. The operator eiY is unitary and, for E ∈ B(H), e−iY EeiY = e−δiY (E) = If |Y |J ≤

1 2

∞ (−i)n (δY )n (E), where δY (E) = [Y, E]. n n=0

then F (Y, E) = e−iY EeiY − E − i[E, Y ] ∈ J

and |F (Y, E)|J ≤ 4|δY (E)|J |Y |J . −iY

−iY

Ae ) = Sp(A) ⊂ Γ and g(e We have Sp(e Using this and (3.4), we obtain that iY

(3.4) iY

−iY

Ae ) = e

g(A)eiY .

|GA (Y ) − GA (0) − (GA )∇ 0 (Y )|J = |g(A + i[A, Y ]) − g(A) − i[g(A), Y ]|J ≤ |g(e−iY AeiY ) − g(A) − i[g(A), Y ]|J + |g(A + i[A, Y ]) − g(e−iY AeiY )|J = |F (Y, g(A))|J + |g(A + i[A, Y ]) − g(e−iY AeiY )|J . By (3.4), A + i[A, Y ]) = e−iY AeiY + F (Y, A) and |F (Y, A)|J ≤ 4|δY (A)|J |Y |J . Let α be a compact in Γ whose interior contains Sp(A). Choose sufficiently small ν > 0 such that A + i[A, Y ] ∈ Ssa (α), for Y ∈ Jsa with |Y |J ≤ ν. Since g is J-Lipschitzian on α, it follows from (3.2) that |g(A + i[A, Y ]) − g(e−iY AeiY )|J = |g(e−iY AeiY + F (Y, A)) − g(e−iY AeiY )|J ≤ D|F (Y, A)|J . Taking (3.4) into account, we complete the proof, since |GA (Y ) − GA (0) − (GA )∇ 0 (Y )|J ≤ |F (Y, g(A))|J + D|F (Y, A)|J ≤ 4|δY (g(A))|J |Y |J + 4D|δY (A)|J |Y |J ≤ 8(g(A) + DA)|Y |2J .

(3.5)

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Let A = A∗ ∈ B(H) and let J be a s. n. ideal. We denote by δA (Jsa ) the closure of δA (Jsa ) = {δA (X) = [A, X] : X ∈ Jsa } in |·|J . Then δA (J) Jsa = iδA (Jsa ) and δA (J)

Jsa = iδA (Jsa ).

(3.6)

1

Let (A) be the commutant of A. If J = S isseparable, then (see [19, Theorem 7.1]) there is a projection ΨA from J on (A) J with ΨA J = 1, and J = ((A) J) + δA (J). (3.7) An operator A is diagonalizable if there is an orthonormal basis in H which consists of eigenvectors of A. If J = S 1 and A = A∗ is a diagonalizable operator then, repeating the argument of [19, Theorem 7.1], we obtain that (3.7) holds for this case also. Corollary 3.3. Let g be a J-Lipschitz function on Γ, where J is J0φ , or J φ , or S, and let A ∈ Ssa (Γ). Then (i) g is Gateaux J-Differentiable at any A ∈ Ssa (Γ) along iδA (Jsa ) and ∇ gA (i[A, Y ]) = i[g(A), Y ];

(ii) if J = S 1 is separable and A ∈ Ssa (Γ) has no eigenvalues, then g is Gateaux J-Differentiable at A along Jsa . Proof. Let α be a compact in Γ whose interior contains Sp(A). Then g is J-Lipschitzian on α. By (3.3), there is D > 0 such that |[g(A), Y ]|J ≤ D|[A, Y ]|J , for Y ∈ J.

(3.8)

For Y ∈ Jsa , set X = i[A, Y ] and T (X) = i[g(A), Y ]. By (3.8), T is a bounded operator from iδA (Jsa ) into J. It follows from (3.2) that g(A + i[A, Y ]) − g(A) ∈ J. By (3.5), there is C > 0 such that ωX (t) = |g(A + tX) − g(A) − tT (X)|J = |g(A + i[A, tY ]) − g(A) − i[g(A), tY ]|J ≤ C|Y |2J |t|2 , (3.9) so that g is Gateaux J-Differentiable at A along iδA (Jsa ) and T coincides with ∇ the restriction of gA to iδA (Jsa ). Extend T by continuity to iδA (Jsa ) and denote the extension also by T . Then T ≤ D. For Z ∈ iδA (Jsa ) and > 0 choose X ∈ iδA (Jsa ) such that . By (3.2), g(A + tZ) − g(A + tX) ∈ J and |X − Z|J ≤ 3D |g(A + tZ) − g(A + tX)|J ≤ D|t(Z − X)|J . Therefore |g(A + tZ) − g(A) − tT (Z)|J ≤ ωX (t) + |g(A + tZ) − g(A + tX)|J + |t||T (Z − X)|J ≤ ωX (t) + D|t(Z − X)|J + D|t||Z − X|J ≤ ωX (t) +

2 |t|. 3

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By (3.9), there is ν > 0 such that ωX (t) ≤ 3 |t| for |t| ≤ ν. Thus ωZ (t) ≤ |t| and g is Gateaux J-Differentiable at A in the direction Z. Part (i) is proved. If J = S 1 is separable and A has no eigenvalues, then (A) J = (0). By (3.7), J = δA (J) and iδA (Jsa ) = Jsa . Davies [9] proved that S p -Lipschitz functions, 1 < p < ∞, can be nondifferentiable. In particular, the function g(t) = |t| is S p -Lipschitzian. Combining this with Corollary 3.3(ii), we see that the Gateaux J-Differentiability of a function at an operator A without eigenvalues does not imply that the function is differentiable on Sp(A) (cf. Proposition 2.4(i)). We consider now some conditions which guarantee the J-Differentiability of a function at any selfadjoint A along (A) Jsa . Lemma 3.4. (i) If g is a continuously differentiable function on Γ then, for S and for every ideal, it is Frechet J-Differentiable at each A ∈ Ssa (Γ) along (A) Jsa . (ii) If g is differentiable on Γ and g is bounded on all compact subsets of Γ, then, for every separable ideal J, the function g is Gateaux J-Differentiable at each A ∈ Ssa (Γ) along (A) Jsa . (iii) Let J be J0φ , J φ , or S and let A ∈ Ssa (Γ) be an operator with finite spectrum. If a continuous function g on Γ is differentiable in all points of Sp(A), then it is Frechet J-Differentiable along (A) Jsa . ∇ ∇ In all cases the G-derivative gA acts by the formula: gA (Z) = g (A)Z. Proof. Choose a compact α in Γ whose interior contains Sp(A). In (i) the derivative g is continuous, in (ii) it is Borel and bounded on α. Hence the operator g (A) is bounded in both cases and the corresponding multiplication operator Φ : Z → g (A)Z is bounded on J: ΦJ = g (A) ≤

sup |g (s)|. s∈Sp(A)

Let > 0 be such that r + s ∈ α, for r ∈ Sp(A) and s ∈ [−, ]. The function 1 [g (r + ts) − g (r)]sdt h(r, s) = g(r + s) − g(r) − g (r)s = 0 is continuous on Sp(A) × [−, ]. Let Z ∈ (A) Jsa and Z ≤ |Z|J ≤ . Then h is continuous on the product of spectra of the commuting operators A and Z. Hence 1 [g (A + tZ) − g (A)]Zdt, h(A, Z) = g(A + Z) − g(A) − g (A)Z = 0

where the integral is the limit of Riemann sums. Since [g (A + tZ) − g (A)]Z ∈ J, we have h(A, Z) ∈ J. Since g is continuous, it is uniformly continuous on α. Therefore g (A + tZ) − g (A) =

sup (r,s)∈Sp(A)×[−,]

|g (r + ts) − g (r)|

≤ sup{|g (r1 ) − g (r2 )| : r1 , r2 ∈ α, |r1 − r2 | < }.

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Hence

|g(A + Z) − g(A) − g (A)Z|J = | ≤ |Z|J

1 0

0

1

IEOT

[g (A + tZ) − g (A)]Zdt |J

[g (A + tZ) − g (A)]dt ≤ |Z|J ω().

Since the module ofcontinuity ω() of g tends to 0 as → 0, (i) is proved. Let Z ∈ (A) Jsa . Then Sp(A + tZ) ⊆ α for t < / Z. Since A and Z commute, all eigenspaces of Z are invariant for A. Observe that if a projection P commutes with A and Z, then P g(A + tZ) − P g(A) = g(P (A + tZ)) − g(P A). Therefore, if P is the projection on Ker(Z), then P (g(A + tZ) − g(A) − tg (A)Z) = g(P (A + tZ)) − g(P A) = 0. Hence without loss of generality, we assume that Ker(Z) = (0). Since all eigenspaces of Z are finite-dimensional, there is an orthonormal basis {en : n = 1, 2, . . .} such that Zen = zn en and Aen = an en (all zn and all an are not necessarily distinct). Let Pn be the projections on (Cen ). Then ω(t) = where Kt =

g(A + tZ) − g(A) − g (A)Z = Kt Z, t

∞ g(an + tzn ) − g(an ) ( − g (an ))Pn . tz n n=1

Since g is bounded on all compact subsets of Γ, g is Lipschitzian on all of them. Hence |g(t) − g(s)| gLip(α) = sup |g(t)| + sup |t − s| t∈α t,s∈α,t=s = sup |g(t)| + sup |g (t)|, t∈α

t∈α

so Kt ≤ 2gLip(α) . To prove (ii), we have to show that |ω(t)|J → 0, as t → 0. The projections QN = N 1 Pn strongly converge to 1. Since J is separable, we have |Z − QN Z|J → 0, as N → ∞ (see [14]). For > 0, choose N such that . Then |Z − QN Z|J ≤ 4g Lip(α) |ω(t)|J ≤ |Kt QN Z|J + |Kt (1 − QN )Z|J ≤

N g(an + tzn ) − g(an ) | − g (an )| + . tzn 2 n=1

Since g is differentiable at every point, we can choose δ such that the first summand is less than 2 for |t| < δ. Hence |ω(t)|J < . Part (ii) is proved. Let us firstly prove (iii) for A = t1, with t ∈ Γ. Given > 0, there is δ > 0 such that t + s ∈ Γ and |g(t + s) − g(t) − g (t)s| ≤ |s|, if |s| ≤ δ.

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Let J = S, X ∈ Ssa and let E(s) be the spectral measure of X. If X ≤ δ, then X g(t1 + X) − g(t1) − g (t1)X = (g(t + s) − g(t) − g (t)s)dE(s) −X

≤

sup |g(t + s) − g(t) − g (t)s| ≤ X.

|s|≤X

Let J = J φ , X ∈ Jsa and |X|J < δ . Then X =

g(t1 + X) − g(t1) − g (t1)X =

∞

∞

i=1 si Pi

and

[g(t + si ) − g(t) − g (t)si ]Pi =

i=1

∞

λi P i ,

i=1

where |λi | ≤ |si |. It follows from [14, III, §§3,4] that |g(t1 + X) − g(t1) − g (t1)X|J ≤ |X|J , so part (iii) is proved for the case whenA = t1. n n n Let A = g(A) = i=1 g(ti )Pi and g (A) = i=1 g (ti )Pi . If i=1 ti Pi . Then n X ∈ (A) Jsa , then X = i=1 Pi XPi . For any > 0, there is δ > 0 such that ti + s ∈ Γ and |g(ti + s) − g(ti ) − g (ti )s| ≤ |s|, if |s| ≤ δ, for all i. Let |X|J ≤ δ. Taking into account the case when A = t1, we have n |g(A+X)−g(A)−g (A)X|J = | Pi (g(ti Pi + Pi XPi ) − g(ti )Pi − g (ti )Pi XPi )Pi |J i=1

≤

n

|g(ti Pi + Pi XPi ) − g(ti )Pi − g (ti )Pi XPi |J ≤

i=1

n

|Pi XPi |J ≤ n|X|J .

i=1

This completes the proof of the lemma. It follows from the proof of Proposition 2.4 that both conditions (i) and (ii) in Lemma 3.4 are also necessary. For A = A∗ ∈ B(H), set EA = (A) + δA (S). By (3.7), (C1) + C(H) = (C1) + S ∞ = S˜∞ ⊆ EA .

(3.10)

Proposition 3.5. Let g be a differentiable J-Lipschitz function on Γ. (i) If J = S 1 is separable, then g is Gateaux J-Differentiable on Γ. 1 (ii) If J = S 1 , then g is Gateaux S 1 -Differentiable along Ssa at any diagonalizable operator A in Ssa (Γ.) (iii) If J = S and g is continuously differentiable, then g is Gateaux S-Dif ∞ ferentiable at any A ∈ Ssa (Γ) along EA Ssa and, in particular, along S˜sa . J Proof. For Jsa ). Let Y ∈ Jsa , let sa ) + ((A) A ∈ Ssa (Γ), set MA = iδA (J J Z ∈ (A) Jsa and X = i[A, Y ] + Z ∈ MA . If J = S, it follows from (3.7) that |Z|J = |ΨA (X)|J ≤ |X|J and |[A, Y ]|J ≤ 2|X|J .

(3.11)

For J = S, considering any conditional expectation Ψ from B(H) onto (A) , we obtain (3.11).

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Set Φ(X) = i[g(A), Y ] + g (A)Z. By (3.8) and (3.11), |Φ(X)|J ≤ |[g(A), Y ]|J + |g (A)Z|J ≤ D|[A, Y ]|J + g (A)|Z|J ≤ 2D|X|J + g (A)|X|J , so that Φ is a linear bounded operator from (MAJ , | · |J ) into J. We have A + tX = A + tZ + it[A, Y ] = A + tZ + it[A + tZ, Y ] − it2 [Z, Y ]. Choose a compact α in Γ whose interior int(α) contains Sp(A) and let > 0 be such that A + tX and A + tZ + it[A + tZ, Y ] belong to Jsa (int(α)) for |t| ≤ . Since g is J-Lipschitzian on α, we obtain from (3.2) that there is D > 0 such that g(A + tX) = g(A + tZ + it[A + tZ, Y ]) + K(t), where |K(t)|J ≤ D|t|2 |[Z, Y ]|J . Therefore |g(A + tX) − g(A) − tΦ(X)|J ≤ |g(A + tZ + it[A + tZ, Y ]) − g(A) − it[g(A), Y ] − g (A)tZ|J + o(|t|) ≤ |g(A + tZ + it[A + tZ, Y ]) − g(A + tZ) − it[g(A + tZ), Y ]|J +|g(A + tZ) − g(A) − g (A)tZ|J + |t[g(A + tZ), Y ] − t[g(A), Y ]|J + o(|t|). Replacing A by A + tZ and Y by tY in (3.5), we have that |g(A + tZ + it[A + tZ, Y ]) − g(A + tZ) − it[g(A + tZ), Y ]|J ≤ 8(g(A + tZ) + DA + tZ)(|tY |J )2 = o(|t|). Since J-Lipschitz functions are Lipschitzian in the usual sense, it follows from Lemma 3.4 that g is Gateaux J-Differentiable at A along (A) Jsa and that ∇ (Z) = g (A)Z. Therefore gA |g(A + tZ) − g(A) − g (A)tZ|J = o(|t|). Finally, since g is J-Lipschitzian on α, we obtain, as above, that |t[g(A + tZ), Y ] − t[g(A), Y ]|J ≤ 2|t||g(A + tZ) − g(A)|J Y ≤ 2D|t|2 |Z|J Y = o(|t|). Combining the above formulae yields |g(A + tX) − g(A) − tΦ(X)|J = o(|t|). ∇ Hence g is Gateaux J-Differentiable at A along (MAJ , | · |J ) and gA (X) = Φ(X). J It follows from (3.6) and (3.7) that MA is dense in Jsa if J = S 1 is separable, or if J = S 1 and Sp(A) is discrete. Hence Φ extends to a bounded operator on Jsa . We also have that MAJ is dense in EA Ssa if J = S. Hence Φ extends to a bounded operator on EA Ssa . Using now the same argument as at the end of Corollary 3.3, we complete the proof.

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It should be noted that, by Proposition 2.4(iii), the continuity of g in Proposition 3.5(iii) is also a necessary condition. Recall that Fsa (Γ) denotes the set of all selfadjoint finite rank operators with ∇ ∇ J we denote the norm of the operator gA which acts from spectrum in Γ. By gA Jsa into J. We obtain now the main result of this section. Theorem 3.6. If J is separable, then the following conditions are equivalent: (i) g is a differentiable, J-Lipschitz function on Γ; (ii) g is Gateaux J-Differentiable on Γ (at any diagonalizable operator A in Ssa (Γ), if J = S 1 ); (iii) g is Gateaux J-Differentiable along Fsa on Ssa (Γ) (at any diagonalizable operator A in Ssa (Γ), if J = S 1 ); (iv) g is Gateaux J-Differentiable on Fsa (Γ) along Fsa and the G-derivative is bounded on any compact subset α of Γ, that is, there exists Kα > 0 such that ∇ J ≤ Kα for all A ∈ Fsa (α). gA Proof. (i) ⇒ (ii) was proved in Proposition 3.5. (ii) ⇒ (iii) is evident. (iii) ⇒ (iv). Clearly, we only have to prove that the G-derivative is bounded on any compact α in Γ. Suppose that there exist α ⊂ Γ, (An ) in Fsa (α) and (Xn ) ∇ (Xn )|J > n. in Fsa such that |Xn |J = 1 and |gA n ∞ ∞ Let H = i=1 Hi where Hi = H. Then A = i=1 Ai is a bounded selfadjoint ˆ n = 0 ⊕ . . . ⊕ 0 ⊕ Xn ⊕ 0 ⊕ . . . are selfadjoint operator on H, Sp(A) ⊆ α, and all X finite rank operators on H. Since the Gateaux J-Differentiability of a function does not depend on the underlying space, g is Gateaux J-Differentiable at A. Hence ∇ ˆ ∇ gA (Xn ) = 0 ⊕ . . . ⊕ 0 ⊕ gA (Xn ) ⊕ 0 ⊕ . . . , for n = 1, . . . , k, . . . . n ∇ ˆ ∇ ˆ n |J = (Xn )|J = |gA (Xn )|J > n and that |X It follows from Remark 1.1 that |gA n ∇ |Xn |J = 1 which contradicts the fact that gA is bounded. (iv) ⇒ (i). By Proposition 2.4(i), g is differentiable on Γ. It follows from [19, Theorem 9.4] that in order to prove that g is J-Lipschitzian it suffices to show that, for any t ∈ Γ, there is a neighbourhood γt of t in Γ such that g is J-Lipschitzian on γt . From this and from Definition 3.1(ii) we derive that we only need to prove that g is J-Lipschitzian on any [a, b] in Γ. Let α = [a, b] ⊂ Γ and A, B ∈ Fsa (α). (If 0 ∈ α then A, B are operators on a finite-dimensional space). The operators G(s) = sA + (1 − s)B belong to Fsa (α), ∇ J ≤ Kα . Since g is Gateaux J-Differentiable along Fsa for s ∈ [0, 1]. Hence gG(s) on Fsa (Γ), it follows from the law of the mean (see [20]) that ∇ |g(A) − g(B)|J ≤ sup gG(s) J |A − B|J ≤ Kα |A − B|J .

(3.12)

s∈[0,1]

Since J is separable, it follows from (3.3) that (3.12) holds for all A, B in Jsa (α). Thus g is J-Lipschitzian on α, so it is J-Lipschitzian on Γ.

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It was proved in [19] that the spaces of S ∞ -Lipschitz and S-Lipschitz functions coincide (we call them Operator Lipschitzian). Let S˜∞ be the unitization of S ∞ . The next result is a refinement of Theorem 3.6 for J = S ∞ . Theorem 3.7. Let g be a function on Γ. The following conditions are equivalent. (i) g is Operator Lipschitzian on Γ; (ii) g is Gateaux S ∞ -Differentiable on Γ; (iii) g is Gateaux S ∞ -Differentiable on Ssa (Γ) along Fsa ; ∞ ∞ (Γ) along Ssa ; (iv) g is Gateaux S ∞ -Differentiable on S˜sa ∞ ∞ ∞ (v) g is S -Differentiable on S˜sa (Γ) along each Z ∈ Ssa . Proof. Operator Lipschitz functions satisfy (3.3) for J = S. It follows from [15, Corollary 3.7 and Theorem 4.1] that every function which satisfies (3.3), for J = S, is differentiable. Therefore Operator Lipschitz functions are differentiable. Thus (i) ⇒ (ii) follows from Theorem 3.6. (ii) ⇒ (iii), (ii) ⇒ (iv) and (iv) ⇒ (v) are evident. (iii) ⇒ (ii) was proved in Theorem 3.6. (v) ⇒ (i). By [19, Theorem 9.4], g is Operator Lipschitzian if it is locally Operator Lipschitzian, that is, if any t in Γ has a neighbourhood where g is Operator Lipschitzian. If g is not Operator Lipschitzian, there is r ∈ Γ such that g is not Operator Lipschitzian in any neighbourhood of r. By Definition 3.1, there is N such that g is not S-Lipschitzian on any segment αn = [r − n1 , r + n1 ] ⊂ Γ, for n ≥ N . We have from (3.1) and the remark after (3.3) that there are finite rank operators An , Bn on Hilbert spaces H(n) with spectra in αn such that g(An ) − g(Bn ≥ nAn − Bn .

(3.13)

Since An , Bn have finite rank, we may assume that H(n) are finite-dimensional. Set ∞ H(n), λn = n−1/2 Bn − An −1 , Añ = An − r1H(n) , H = n=N

A=

∞ n=N

An = r1H +

∞ n=N

Añ and X =

∞

λn (Bn − An ).

n=N

Since Sp(Añ ) ⊆ [− n1 , n1 ], we have Añ ≤ n−1 and Sp(A) ⊆ Γ. Hence A ∈ ∞ S˜sa (Γ). Since λn (Bn − An ) ≤ n−1/2 , the operator X is compact. Since g is ∞ ∞ ˆ ∈ Ssa (see (2.1)) such that S -Differentiable at A along X, there are δ > 0 and X ˆ = o(|t|) for t ∈ (−δ, δ). g(A + tX) − g(A) − tX Hence there is D > 0 such that g(A + tX) − g(A) ≤ D|t| for t ∈ (−δ, δ), so that g(An + tλn (Bn − An )) − g(An ) ≤ D|t| for all n.

(3.14)

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We have Sp(An − Bn ) ⊆ [− n2 , n2 ]. Therefore An − Bn ≤ n2 and we have that 1/2 Bn − An ≤ 2n−1/2 → 0. Substituting λ−1 λ−1 n = n n for t in (3.14), we obtain 1/2 Bn − An g(Bn ) − g(An ) ≤ Dλ−1 n ≤ Dn

which contradicts (3.13). Remark 3.8. (i) It follows from Theorems 3.6 and 3.7 that in the infinite dimensional case any Gateaux J-Differentiable function is automatically J-Lipschitzian and, therefore, Lipschitzian in the usual sense. We will see in Section 5 that this is not true if dim H < ∞. Moreover, in the finite-dimensional case parts (i) and (iv) in Theorem 3.6 are equivalent to each other and not equivalent to part (ii). (ii) If g is continuously differentiable, then the conditions of Theorem 3.7 are equivalent to the condition: (vi) g is Gateaux S-Differentiable at each A ∈ Ssa (Γ) along EA Ssa . Indeed, (i)⇒ (vi) was proved in Proposition 3.5. Conversely, if (vi) holds, then 1H ∈ EA Ssa (see (3.10)) and, by Proposition 2.4(iii), g is continuously differentiable on Γ. We obtain from (3.7) that S ∞ ⊆ EA for any A = A∗ . Hence (vi) implies (ii) of Theorem 3.7. (iii) From the proof of the implication (v) ⇒ (i) we see that a function g is S-Lipschitzian on Γ if it satisfies the following “Gateaux” type condition: ∞ ∞ (Γ) and X ∈ Ssa . g(A + tX) − g(A) = O(|t|), for A ∈ S˜sa It is formally much weaker than “Frechet” type condition in Definition 3.1. Problem 3.9. By Theorem 3.7, the Gateaux S-Differentiability of g on Ssa (Γ) along ∞ ∞ ∞ Ssa is equivalent to its Gateaux S-Differentiability on S˜sa (Γ) along Ssa . Are the conditions of Theorem 3.6 equivalent to the condition: (v) g is Gateaux J-Differentiable on J˜s (Γ) along Jsa ? Corollary 3.10. Let g be a Gateaux J-Differentiable function on Γ. ∗ (i) Let J = J0φ = S ∞ , let φ∗ be the adjoint function and let I = J0φ be the corresponding separable s. n. ideal. Then g is Gateaux I-Differentiable on Γ. (ii) Let J = S p , p ∈ (1, ∞), and let p = p/(p − 1). Then g is Gateaux q S -Differentiable on Γ, for min(p, p ) ≤ q ≤ max(p, p ). (iii) If J = S 1 , then g is Gateaux I-Differentiable on Γ for any separable ideal I. (iv) Let J = S ∞ . Then g is Gateaux I-Differentiable on Γ, for any separable ideal I = S 1 , and Gateaux S 1 -Differentiable at any diagonalizable operator A in 1 Ssa (Γ) along Ssa . Proof. If g is a J0φ -Lipschitz function, it follows from [19, Proposition 4.6 and ∗ Corollary 5.4] that it is J0φ -Lipschitzian. If g is a S p -Lipschitz function, p ∈ (1, ∞), then it follows from [19, Proposition 4.6 and Corollary 6.5] that it is also S q Lipschitzian for min(p, p ) ≤ q ≤ max(p, p ). If g is a S 1 - or S ∞ -Lipschitz function, it follows from [19, Proposition 4.6 and Corollary 6.3] that it is I-Lipschitzian, for

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any separable s. n. ideal I. Combining this with Theorem 3.6, we complete the proof. ∇ Finally, we consider briefly the form of the G-derivative gA . If g is Gateaux J-Differentiable on Γ, then g is differentiable and we define a function gˇ(t, s) on Γ × Γ by

gˇ(t, s) = (g(t) − g(s))/(t − s), if t = s, and gˇ(t, t) = g (t). For sufficiently smooth functions g, the G-derivative can be expressed via the spectral measure E(s) of A (see [8, 4, 23]) in the form of the double operator integral ∇ (X) = gˇ(t, s)dE(t)XdE(s). gA If Sp(A) is a finite set then this formula can be easily obtained for any Gateaux J-Differentiable function (we will do it below for the reader’s convenience). Let Sp(A)= (λ1 , . . . , λn ) and P (λi ) = E((λi )). With respect to the decomposition H = ni=1 P (λi )H, every operator X on H has the block-matrix form X = (Xij ), 1 ≤ i, j ≤ n. The n × n matrix M (A, g) = (mij ), where mij = gˇ(λi , λj ),

(3.15)

defines an operator (an Hadamard multiplier) on B(H) by the formula: M (A, g) ◦ X = (mij Xij ).

(3.16)

Proposition 3.11. Let J be J0φ , J φ or S. Let g be a Gateaux J-Differentiable function on Γ, let A ∈ Ssa (Γ) and Sp(A) = (λ1 , . . . , λn ) be a finite set. For X ∈ Jsa , ∇ (X) = M (A, g) ◦ X = gA

n

gˇ(λi , λj )P (λi )XP (λj ).

(3.17)

i,j=1

Proof. The second equality in (3.17) is evident, so we only need to prove the first one. By Theorems 3.6 and 3.7, g is differentiable on Γ and J-Lipschitzian. If Y = (Yij ) ∈ (A) Jsa , then Yij = 0 for i = j. Since g (A) = ni=1 g (λi )P (λi ), ∇ (Y ) = g (A)Y = M (A, g) ◦ Y . we have from Lemma 3.4(iii) that gA Let Z = (Zij ) ∈ Jsa , and set [A, Z] = (Kij ) and [g(A), Z] = (Bij ). Then Kij = (λi − λj )Zij and Bij = (g(λi ) − g(λj ))Zij , so that Bij = mij Kij . By Corollary 3.3, ∇ (i[A, Z]) = i[g(A), Z] = M (A, g) ◦ i[A, Z]. gA

Let X ∈ Jsa . It follows from (3.7) that X = Y + i[A, Z], where Y ∈ (A) ∇ and Z ∈ Jsa . Hence gA (X) = M (A, g) ◦ X.

Jsa

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4. Frechet J-Differentiable functions We saw in Proposition 2.4 that all Frechet J-Differentiable functions are continuously differentiable. On the other hand, by Theorem 3.6, for any separable J, Gateaux J-Differentiable functions are just differentiable, J-Lipschitz functions. It was proved in [19] that Gateaux J-Differentiable functions can have discontinuous derivatives. Hence the notions of the Frechet and Gateaux J-Differentiability do not coincide for all separable ideals. For S = B(H) the situation is different: we will establish that the spaces of Gateaux and Frechet S-Differentiable functions coincide. First we will show that the Frechet J-Differentiability of a function is equivalent to its Frechet J-Differentiability along the set Fsa of selfadjoint finite rank operators. The proof can be extended to a more general setting of Frechet Differentiable maps on Banach spaces but we will preserve our operator framework. Theorem 4.1. Let J be a separable ideal or S and let g be a continuous function on Γ. If g is Frechet J-Differentiable at A ∈ Ssa (Γ) along Fsa , then it is Frechet J-Differentiable at A along Jsa . Proof. Let α be a compact in Γ such that Sp(A) ⊂ int(α). There is ρ > 0 such that A + Z ∈ Ssa ( int(α)), for any Z ∈ Jsa with |Z|J < ρ. Let T be the G-derivative of g at A along Fsa . For any > 0, there is 0 < δ() < ρ such that |g(A + X) − g(A) − T (X)|J ≤ |X|J ,

(4.1)

for all X ∈ Fsa with |X|J ≤ δ(). 1. Let J = S ∞ be separable. Since T is a bounded operator from Fsa into J and Fsa is dense in Jsa , T extends to a bounded operator from Jsa into J which we also denote by T . Let X ∈ Jsa and |X|J ≤ δ(). There exist Xn in Fsa such that |Xn |J ≤ |X|J and |X − Xn |J → 0. Then |T (X) − T (Xn )|J → 0, g(A + Xn ) − g(A) − T (Xn ) ∈ J and, for any n, |g(A + Xn ) − g(A) − T (Xn )|J ≤ |Xn |J ≤ |X|J . We also have that X −Xn ≤ |X −Xn |J → 0, so that g(A+X)−g(A+Xn ) → 0. Hence (g(A + X) − g(A) − T (X)) − (g(A + Xn ) − g(A) − T (Xn )) ≤ g(A + X) − g(A + Xn + T (X) − T (Xn ) ≤ g(A + X) − g(A + Xn ) + |T (X) − T (Xn )|J → 0, as n → ∞. By [14, Theorem III.5.1], if a sequence of operators Sn from J converge to S ∈ B(H) in the weak operator topology and |Sn |J ≤ M , for some M > 0, then S ∈ J and |S|J ≤ M . This implies that g(A + X) − g(A) − T (X) ∈ J and |g(A + X) − g(A) − T (X)|J ≤ |X|J .

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Thus g is Frechet J-Differentiable at A along Jsa . 2. Let J be S ∞ or S. If P is a polynomial then P (A + K) − P (A) ∈ S ∞ , for any K ∈ S ∞ , and the map X → P (X) is continuous in the strong operator topology on Ssa (α). Choosing polynomials uniformly converging to g on α, we obtain that a) g(A + K) − g(A) ∈ S ∞ for any K ∈ S ∞ , b) the map X → g(X) is continuous in the strong operator topology on Ssa (α). It follows from a) that, for any X ∈ Fsa , 1 T (X) = lim [g(A + tX) − g(A)] ∈ S ∞ . t→0 t ∞ Thus T (Fsa ) ⊆ S . Since T is bounded, it extends to a bounded operator on S ∞ which we also denote by T . Hence T ∗∗ is a bounded operator on S = B(H). Let X ∈ Ssa and X ≤ δ(), so that (4.1) holds. By Kaplansky density theorem there exists a directed set (Xλ )λ∈Λ of operators in Fsa , converging to X in the strong operator topology, such that Xλ ≤ X. By (4.1), for all λ, g(A + Xλ ) − g(A) − T (Xλ ) ≤ Xλ ≤ X.

(4.2)

Let R ∈ S 1 and FR (Z) = T r(RZ) be the corresponding functional on S. Then FR (T (Xλ )) = (T ∗ FR )(Xλ ) → (T ∗ FR )(X) = FR (T ∗∗ (X)). Hence T (Xλ ) converge to T ∗∗ (X) in the σ(S, S 1 ) topology. Since, by b), g(A+Xλ ) converge to g(A + X) in the strong operator topology, we obtain that g(A + Xλ ) − g(A) − T (Xλ ) converge to g(A + X) − g(A) − T ∗∗ (X) in the σ(S, S 1 ) topology. It is well known that the norm on B(H) is σ(S, S 1 )− semi-continuous from below. Therefore, by (4.2) g(A + X) − g(A) − T ∗∗ (X) ≤ limg(A + Xλ ) − g(A) − T (Xλ ) ≤ X. Theorem 4.1 implies in particular that Frechet S ∞ -Differentiable functions are also Frechet S-Differentiable. We proceed now to prove that a function is Frechet S-Differentiable on Γ if and only if it is Gateaux S-Differentiable on Γ. We start with the following simple result. Lemma 4.2. Let g be a continuous function on Γ, let J be a s. n. ideal or S and let ∞ ∞ ∞ Hi , A = Ai ∈ Ssa (Γ) and X = Xi ∈ J(H), H= i=1

i=1

i=1

where Ai , Xi are bounded selfadjoint operators on Hilbert spaces Hi . If g is J-Differentiable at A along X, then it is J-differentiable at each Ai along Xi and ∇ gA (X) =

∞ i=1

∇ gA (Xi ). i

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Proof. Let Qi be the projections on Hi and set 1 1 B(t) = [g(A + tX) − g(A)] and Bi (t) = [g(Ai + tXi ) − g(Ai )]. t t ∞ ∇ ∇ Then B(t) = i=1 Bi (t) → gA (X) in |·|J , so that |Qi B(t)Qj −Qi gA (X)Qj |J → 0, as t → 0. Hence ∇ ∇ (X)Qi and 0 = Qi B(t)Qj → Qi gA (X)Qj , if i = j, Bi (t) = Qi B(t)Qi → Qi gA

in | · |J . Thus g is J-differentiable at each Ai along Xi , ∇ (Xi ) gA i

=

∇ Qi gA (X)Qi

and

∇ gA (X)

=

∞

∇ gA (Xi ). i

i=1

If all Hi = H and Ai = B, we write H ∞ for H and B ∞ for A. An operator X in B(H) has infinite multiplicity if it is unitarily equivalent to the operator B ∞ for some B ∈ Ssa . We denote by Φ the set of all selfadjoint operators of infinite multiplicity and by Φ(Γ) the set of all operators from Φ with spectrum in Γ. Theorem 4.3. Let g be a function on Γ. The following conditions are equivalent: (i) g is Gateuax S-differentiable on Γ; (ii) g is Gateuax S-differentiable on Φ(Γ) along Ssa ; (iii) g is Frechet S-differentiable on Γ; (iv) g is Frechet S-differentiable on Φ(Γ) along Ssa ; (v) g is Frechet S ∞ -differentiable on Γ; (vi) g is Frechet S-differentiable on Ssa (Γ) along Fsa . Proof. Clearly, (i) ⇒ (ii), (iii) ⇒ (iv), (iii) ⇒ (v), (v) ⇒ (vi), (iii) ⇒ (i) and (iv) ⇒ (ii) are evident, (vi) ⇒ (iii) follows from Theorem 4.1. (ii) ⇒ (iii). Let A ∈ Ssa (Γ). Then Sp(A∞ ) ⊂ Γ and, by (ii), g is Gateaux S-differentiable at A∞ along Ssa (H ∞ ) = {X = X ∗ ∈ B(H ∞ )}. It follows from Lemma 4.2 that g is Gateaux S-differentiable at A along Ssa (H). If g is not Frechet S-differentiable at A along Ssa (H) then there exist > 0, Xn ∈ Ssa (H) with Xn = 1, and tn → 0 such that 1 ∇ [g(A + tn Xn ) − g(A)] − gA (Xn ) > . (4.3) tn ∞ Consider X = i=1 Xi ∈ Ssa (H ∞ ). Taking into account Lemma 4.2 and the fact that g is Gateaux S-differentiable at A∞ along Ssa (H ∞ ), we obtain that 1 1 ∇ ∇ [g(A∞ + tX) − g(A)] − gA ∞ (X) = sup [g(A + tXi ) − g(A)] − gA (Xi ) → 0, t t i as t → 0. This contradicts (4.3), so g is Frechet S-differentiable at A along Ssa (H). Let J be a s. n. ideal or S. We denote by F DJ (Γ) the spaces of all Frechet and by GDJ (Γ) the space of all Gateaux J-Differentiable functions on Γ. The space of all J-Lipschitz functions on Γ we denote by OLJ (Γ). We call S-Lipschitz

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functions Operator Lipschitzian and, taking Theorem 4.3 into account, we call Gateaux (Frechet) S-Differentiable functions Operator Differentiable. It should be noted that though the proof of Theorem 4.3 is only valid for infinite-dimensional spaces, we will show in Theorem 5.7 that the result also holds for finite-dimensional spaces. We shall now summarize various results about the spaces of J-Lipschitz, Gateaux and Frechet J-Differentiable functions. Proposition 4.4. (i) Let J be J0φ , J φ or S and let Γ ⊆ R. Then F DJ (Γ) ⊆ GDJ (Γ) ⊆ OLJ (Γ). (ii) If J is separable, then F DJ (Γ) ⊂ GDJ (Γ), and their difference contains differentiable but not continuously differentiable functions. (iii) For 1 < p < ∞, GDS p (Γ) ⊂ OLS p (Γ), and their difference contains Lipschitz non-differentiable functions. (iv) F DS (Γ) = GDS (Γ) ⊂ OLS (Γ) = OLS ∞ (Γ) = GDS ∞ (Γ), and the difference between OLS (Γ) and GDS (Γ) contains differentiable but not continuously differentiable functions. Proof. For all ideals, F DJ (Γ) ⊆ GDJ (Γ). If J is separable (J = J0φ ), then, by Theorem 3.6, GDJ (Γ) ⊆ OLJ (Γ). Let J be J φ or S and let J0 be J0φ or S ∞ , respectively. Then GDJ (Γ) ⊆ GDJ0 (Γ) and, by the above, GDJ0 (Γ) ⊆ OLJ0 (Γ). It was proved in [19] that OLJ0 (Γ) = OLJ (Γ). This proves (i). Let J be separable. It was established in [19] that the function 1 (4.4) g(t) = t2 sin( ), for t = 0, and g(0) = 0 t is S-Lipschitzian and, hence, J-Lipschitzian on R. Since g is differentiable, we have from Theorem 3.6 that g ∈ GDJ (Γ). However, since g is not continuously differentiable, by Proposition 2.4, g ∈ F DJ (Γ). Part (ii) is proved. By Theorem 3.6, GDS p (Γ), for 1 < p < ∞, consists of all differentiable, S p -Lipschitz functions on Γ. On the other hand, Davies [9] showed that OLS p (Γ) contains non-differentiable functions. Part (iii) is proved. It was proved in [19] that OLS (Γ) = OLS ∞ (Γ). By Theorem 3.7, OLS ∞ (Γ) = GDS ∞ (Γ). It follows from Proposition 2.4 and Theorem 4.3 that F DS (Γ) = GDS (Γ) and consists of continuously differentiable functions. Johnson and Williams [15] established that S-Lipschitz functions are differentiable. Thus in order to prove (iv) we only need to show that OLS (Γ) = GDS (Γ). The function g in (4.4) lies in OLS (Γ) and is not continuously differentiable. Hence g ∈ GDS (Γ). Problem 4.5. (i) (cf. Corollary 3.10) Does the Frechet Differentiability have “interpolation” properties? 1) Let g be a Frechet S p -Differentiable function on Γ, p ∈ (1, ∞), and let p p = p−1 . Is g Frechet S q -Differentiable, for min(p, p ) ≤ q ≤ max(p, p )? 2) Are Frechet S 1 -Differentiable functions Operator Differentiable? 3) Are Operator Differentiable functions Frechet S p -Differentiable?

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(ii) Let J be a separable ideal or S and let g be a continuously differentiable, J-Lipschitz function on Γ. Is g Frechet J-Differentiable on Γ? In particular, are all continuously differentiable functions on Γ Frechet S 2 -Differentiable? In the next section we will see that Frechet J-Differentiable functions are characterized by continuity of their G-derivatives.

5. Continuity of the G-derivative In this section we prove that Frechet J-Differentiable functions have continuous G-derivative and, conversely, that Gateaux J-Differentiable function with continuous G-derivative are Frechet J-Differentiable. We also consider functions acting on operators on finite-dimensional spaces. We show that in this case Gateaux and Frechet Differentiable functions are just the functions differentiable in the usual sense. Moreover, the Gateaux derivatives of such functions are continuous if and only if their usual derivatives are continuous. Lemma 5.1. Let J be a s. n. ideal or S, let g be a Frechet J-Differentiable function on Γ and let α be a compact in Γ. Then, for any > 0, there exists δ > 0 such that, for any X ∈ Jsa with |X|J ≤ δ, ∇ |g(A + X) − g(A) − gA (X)|J ≤ |X|J for A ∈ Ssa (α).

Proof. By contradiction. Let An ∈ Ssa (α) and let Xn ∈ Jsa be such that |Xn |J → 0 and, for some > 0, ∇ |g(An + Xn ) − g(An ) − gA (Xn )|J > |Xn |J . (5.1) n ∞ ∞ ∞ Let H ∞ = n=1 Hn , where all Hn = H. Then A = n=1 An ∈ B(H ) with Sp(A) ⊆ α, and g is Frechet J-differentiable at A along Jsa . Hence there is δ > 0 such that for all Y ∈ Jsa (H ∞ ) with |Y |J ≤ δ ∇ (Y )|J ≤ |Y |J . |g(A + Y ) − g(A) − gA

Set Yn = 0 ⊕ . . . ⊕ 0 ⊕ Xn ⊕ 0 ⊕ . . . . Then |Yn |J = |Xn |J → 0 and, by Lemma 4.2 and (5.1), ∇ ∇ (Yn )|J = |g(An + Xn ) − g(An ) − gA (Xn )|J > |Yn |J . |g(A + Yn ) − g(A) − gA n

This contradiction proves the lemma. Jsa

∇ ∇ Recall that by gA J we denote the norm of the operator gA which acts from into J.

Theorem 5.2. Let J be a s. n. ideal or S, let g be a Frechet J-Differentiable function on Γ and let α ⊂ Γ. Then, for any ν > 0, there exists ρ > 0 such that ∇ ∇ A, B ∈ Ssa (α), B − A ∈ J and |B − A|J ≤ ρ imply gA − gB J ≤ ν.

(5.2)

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Proof. Set = ν4 . By Lemma 5.1, there exists δ > 0 such that, for any X ∈ Jsa with |X|J ≤ δ and any R ∈ Ssa (α), ∇ |g(R + X) − g(R) − gR (X)|J ≤ |X|J .

Set ρ = 2δ . Let A, B ∈ Ssa (α) be such that |B − A|J ≤ ρ. Let X ∈ Jsa with |X|J = ρ. Then |B − A + X|J ≤ 2ρ = δ. Therefore ∇ |g(B + X) − g(B) − gB (X)|J ≤ |X|J , ∇ (B − A)|J ≤ |B − A|J , |g(B) − g(A) − gA ∇ (B − A + X)|J ≤ |B − A + X|J . |g(B + X) − g(A) − gA ∇ ∇ ∇ (B − A + X) = gA (B − A) + gA (X), we have Since gA ∇ ∇ ∇ |gB (X) − gA (X)|J ≤ |gB (X) − g(B + X) + g(B)|J ∇ ∇ ∇ +|g(B + X) − g(A) − gA (B − A) − gA (X)|J + |gA (B − A) − g(B) + g(A)|J ≤ |X|J + |B − A + X|J + |B − A|J ≤ 4ρ = νρ. ∇ ∇ − gA J ≤ ν. Since this holds for all X ∈ Jsa with |X|J = ρ, we have gB

Corollary 5.3. A Gateaux J-Differentiable function g on Γ is Frechet J-Differenti∇ is uniformly continuous in · J for each able if and only if its G-derivative gA compact α in Γ, that is, if (5.2) holds. Proof. If g is Frechet J-Differentiable, then (5.2) follows from Theorem 5.2. Con∇ versely, set Φ(B) = gB for B ∈ Ssa (Γ). Let A ∈ Ssa (Γ) and let α be a compact in Γ such that Sp(A) ⊂ int(α). There exists d > 0 such that A + X ∈ Ssa (α) for all X in Jsa with |X|J ≤ d. It follows from the general Newton-Leibniz formula for maps with continuous G-derivative that, for any such X, 1 g(A + X) − g(A) = Φ(A + tX)(X)dt. 0

Therefore

|g(A + X) − g(A) − Φ(A)(X)|J ≤

0

1

Φ(A + tX) − Φ(A)J |X|J dt.

Since Φ satisfies (5.2), for any > 0 there exists δ ≤ d such that |X|J < δ implies Φ(A + tX) − Φ(A)J < . Hence |g(A + X) − g(A) − Φ(A)X|J ≤ |X|J . Thus g is Frechet J-Differentiable on Γ. In the proofs of Theorem 5.2 and Corollary 5.3 we used the fact that H is infinite-dimensional. The case when dim H < ∞ will be considered at the end of the section. It follows from (3.10) that for A = A∗ ∈ B(H), EA = EA(r) , for r ∈ R, where A(r) = A − r1H . If g is a continuously differentiable, Operator Lipschitz (S-Lipschitz) function on Γ then, by Proposition 3.5, g is Gateaux S-Differentiable at each A ∈ Ssa (Γ) along

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EA Ssa . We modify now the proofs of Lemma 5.1 and Theorem 5.2 to prove the ∇ in r. strong continuity of the G-derivative gA(r) Proposition 5.4. Let g be a continuously differentiable, Operator Lipschitz function on Γ, letA ∈ Ssa (Γ) and let d > 0 be such that A(r) ∈ Ssa (Γ) for r ∈ [−d, d]. Let X ∈ EA Ssa . (i) For any > 0, there exists δ > 0 such that, for all t ∈ [−δ, δ], ∇ g(A(r) + tX) − g(A(r)) − tgA(r) (X) ≤ |t|, for r ∈ [−d, d]. ∇ (X) is uniformly continuous on [−d, d] : for (ii) The function X(r) = gA(r) any > 0, there exists δ > 0 such that X(s) − X(r) ≤ for all s, r ∈ [−d, d] with |s − r| ≤ δ.

Proof. Suppose that there exist > 0, rn ∈ [−d, d] and tn → 0 such that ∇ g(A(rn ) + tn X) − g(A(rn )) − tn gA(r (X) > |tn |. (5.3) n) ∞ ∞ Then B = n=1 A(rn ) ∈Ssa (Γ). Set Y = n=1 Xn with all Xn = X. It follows from (3.10) that Y ∈ EB Ssa . By Theorem 3.7, g is S-Differentiable at B along Y . Hence there exists δ > 0 such that |t| ≤ δ implies ∇ (Y ) ≤ |t|. g(B + tY ) − g(B) − tgB

From this and from Lemma 4.2 it follows that ∇ sup g(A(rn ) + tX) − g(A(rn )) − tgA(r (X) ≤ |t|. n) n

which contradicts (5.3). This contradiction proves part (i). It follows from (i) that X(r) is the uniform limit of the functions Xt (r) =

g(A(r) + tX) − g(A(r)) , t

that is, supr∈[−d,d] X(r) − Xt (r) → 0, as t → 0. Since all Xt (r) are continuous in r on [−d, d], the function X(r) is also continuous on [−d, d]. Therefore it is uniformly continuous on [−d, d]. In the rest of the section we study Gateaux and Frechet J-Differentiable functions in the case when H is finite-dimensional. Since all ideals of S = B(H) coincide with S and all norms are equivalent, we can speak simply about Gateaux and Frechet Differentiable functions. We denote by D1 (Γ) the space of all differentiable in the usual sense functions on Γ and by C (1) (Γ) the space of all continuously differentiable functions on Γ. It is well known (see, for example, [13, Problem VI.140] ) that functions from C (1) (Γ) are Frechet Differentiable and their Gateaux derivatives can be written in the form (3.17), if dim H < ∞. We will see now that the same is true for all functions from D1 (Γ). We will prove now the following auxiliary result.

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Lemma 5.5. Let dim H < ∞ and let A = A∗ ∈ B(H). There are > 0 and C > 0 such that, for any T = T ∗ ∈ B(H) with T − A < , there exists a unitary U = U (T ) such that (i) U ∗ T U commutes with A, (ii) U ∗ T U − A ≤ C T − A , (iii) U − 1 ≤ CT − A. Proof. Let (λ1 , . . . , λk ) be distinct eigenvalues of A and let Pi be the projections on the corresponding eigenspaces Hi of A. It follows from Theorem I.3.1 of [14] that there are numbers ρ > 0 and C1 , . . . , Ck > 0 such that T − A < ρ implies that there are mutually orthogonal projections Q1 (T ), . . . , Qk (T ) (Riesz projections) on k subspaces Ki (T ) invariant for T such that i=1 Qi (T ) = 1 and Pi − Qi (T ) < Ci T − A, for i = 1, . . . , k. If Ci T − A < 1 then dim Hi = dim Ki (T ) and there are partial isometries Ui from Hi onto Ki (T ) such that Ui − Pi < Di T − A, k for some constants Di > 0 (see, for example, [25, §105]). Set U = i=1 Ui . Then U is a unitary operator on H, U − 1 =

k

(Ui − Pi ) ≤ DT − A,

i=1

where D =

k i=1

Di , and

U ∗ T U − A ≤ U ∗ T U − T U + T U − T + T − A ≤ U ∗ − 1T U + T U − 1 + T − A ≤ (2T D + 1)T − A ≤ (3AD + 1)T − A. If h ∈ Hi , then U ∗ T U h = Ui∗ T Ui h ∈ Hi . Hence A(U ∗ T U )h = λi (U ∗ T U )h = (U T U )Ah. Thus A commutes with U ∗ T U . ∗

Theorem 5.6. Let dim H < ∞, let S = B(H) and A = A∗ ∈ Ssa (Γ). If a continuous function g on Γ is differentiable in all points of Sp(A), then the map T → g(T ) is Frechet Differentiable at A. Proof. Let Sp(A) = (λ1 , . . . , λn ) and let M = M (A, g) be the matrix considered in (3.15) which defines a Hadamard multiplier on B(H) (see (3.16)). First we consider two particular cases. 1. Let T ∈ Ssa (Γ) commute with A. By Lemma 3.4(iii), g is Frechet Operator Differentiable along T and ∇ g(T ) − g(A) − gA (T − A) = g(T ) − g(A) − g (A)(T − A)

= g(T ) − g(A) − M ◦ (T − A) = o(T − A).

(5.4)

2. Let T = U AU ∗ , where U is a unitary operator, and let p(t) be a polynomial such that g(λi ) = p(λi ) and g (λi ) = p (λi ), for i = 1, . . . , k. Then g(A) = p(A)

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and, by (3.15), M (A, g) = M (A, p). Since f (U AU ∗ ) = U f (A)U ∗ , for any continuous function f , we have g(U AU ∗ ) = p(U AU ∗ ). Since any polynomial is Frechet Operator Differentiable, it follows from (3.17) that g(U AU ∗ ) − g(A) − M (A, g) ◦ (U AU ∗ − A) (5.5) = p(U AU ∗ ) − p(A) − M (A, p) ◦ (U AU ∗ − A) = o(U AU ∗ − A). 3. Finally, we consider the general case. Let the numbers , C > 0 and the unitary operator U = U (T ) be the same as in Lemma 5.5. Set B = U ∗ T U . Since B commutes with A, we obtain from (5.4) and (5.5) g(T ) − g(A) − M ◦ (T − A) = g(U BU ∗ ) − g(A) − M ◦ (U BU ∗ − A) ≤ g(U BU ∗ ) − g(U AU ∗ ) − U [M ◦ (B − A)]U ∗ +g(U AU ∗ ) − g(A) − M ◦ (U AU ∗ − A) +U [M ◦ (B − A)]U ∗ + M ◦ (U AU ∗ − A) − M ◦ (U BU ∗ − A) ≤ o(B − A) + o(U AU ∗ − A) + U [M ◦ (B − A)]U ∗ + M ◦ [U (A − B)U ∗ ]. Since U AU ∗ − A ≤ 2U − 1A, it follows from Lemma 5.5 that o(B − A) + o(U AU ∗ − A) = o(T − A). The operator X → M ◦X is bounded: M ◦ X ≤ M X. Therefore, by Lemma 5.5, U [M ◦ (B − A)]U ∗ + M ◦ [U (A − B)U ∗ ] ≤ U [M ◦ (B − A)]U ∗ − M ◦ (B − A) + M ◦ [U (A − B)U ∗ − (A − B)] ≤ 2U − 1M ◦ (B − A) + M U (A − B)U ∗ − (A − B) ≤ 2U − 1M B − A + 2M U − 1A − B ≤ 4M C 2 T − A2 . Hence g(T )−g(A)−M ◦(T −A) = o(T −A), so that g is Frechet Differentiable at A. Corollary 5.7. Let dim H < ∞, let S = B(H) and let g be a continuous function on Γ. The map A → g(A) is Gateaux Differentiable at each A ∈ Ssa (Γ) if and only if it is Frechet Differentiable at each A ∈ Ssa (Γ), and if and only if g ∈ D1 (Γ). ∇ ∗ For unitary U , we have g(U AU ∗ ) = U g(A)U ∗ . Hence gUAU ∗ (U XU ) = ∗ for X = X . Therefore

∇ U gA (X)U ∗

∇ ∇ ∇ ∗ ∗ ∇ gUAU − gA (X) ∗ (X) − gA (X) = U gA (U XU )U ∇ ∇ ∇ ∇ ≤ U − 1gA (U ∗ XU )U ∗ + (gA (U ∗ XU ) − gA (X))U ∗ + gA (X)U ∗ − 1 ∇ ≤ 4gA X1 − U , so that ∇ ∇ ∇ (5.6) gUAU ∗ − gA ≤ 4gA 1 − U . ∇ is uniformly continProposition 5.8. Let dim H = n < ∞. The G-derivative gA uous with respect to A on any compact α in Γ if and only if g is continuously differentiable.

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∇ Proof. If gA is continuous, we have from (3.15) that g is continuous. Conversely, let g be continuous and α ⊂ Γ. Let numbers , C > 0 be as in Lemma 5.5 and let T − A < , for T, A ∈ Ssa ( int(α)). Set B = U ∗ T U , where U = U (T ) is the same as in Lemma 5.5. Then B commutes with A and, by (5.6), ∇ ∇ ∇ ∇ ∇ ∇ ∇ ∇ gA − gT∇ ≤ gA − gB + gU ∗ T U − gT ≤ gA − gB + 4gT 1 − U .

There is a basis (ei ) in H (it depends on B) with respect to which A = diag(λ1 , . . . , λn ), B = diag(µ1 , . . . , µn ) and |λi − µi | ≤ A − B , i = 1, . . . , n. It ∇ ∇ − gB = M (A, g) − M (B, g) , where M (A, g) and follows from (3.17) that gA M (B, g) are considered with respect to the basis (ei ). Since g is continuous, we obtain from (3.15) that there is R > 0 such that, for T ∈ Ssa (int(α)), gT∇ ≤ R and M (A, g) − M (B, g) ≤ R max{|λi − µi |} ≤ RA − B. ∇ Hence gA − gT∇ ≤ R A − B + 4R 1 − U . Therefore, by Lemma 5.5, ∇ − gT∇ ≤ RCT − A + 4RCT − A, gA ∇ ∇ is continuous. Since the set Ssa (α) is compact, gA is uniformly continso that gA uous.

6. G-Derivative of the convolution of functions In this section we obtain a formula for the G-derivative of the convolution of two functions which will later be used for further work on approximation problems. Let g be a continuously differentiable function on Γ and let ϕ be a continuous function on R with compact support. Their convolution ∞ g(t − r)ϕ(r)dr (g ∗ ϕ)(t) = −∞

is continuously differentiable and (g ∗ ϕ) = g ∗ ϕ. We will prove now analogues of this formula for Operator Lipschitz and Operator Differentiable functions. Let P (t) = tn . Then ∞ ∞ n n i n−i (t − r) ϕ(r)dr = Cn t r i ϕ(r)dr, (6.1) (P ∗ ϕ)(t) = −∞

where

−∞

i=0

Cni

are the binomial coefficients. If B is a selfadjoint bounded operator, then ∞ ∞ n Cni B n−i r i ϕ(r)dr = (B − r1H )n ϕ(r)dr. (P ∗ ϕ)(B) = i=0

Therefore, for any polynomial P , (P ∗ ϕ)(B) =

−∞

∞

−∞

−∞

P (B − r1H )ϕ(r)dr.

Let ρ > 0 and let compacts β ⊂ γ in Γ be such that supp(ϕ) ⊆ [−ρ, ρ] and that Sp(B + X) ⊆ γ, for B ∈ Ssa (β) and X = X ∗ with X ≤ ρ. If polynomials

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Pn uniformly converge to g on γ , then Pn ∗ ϕ uniformly converge to g ∗ ϕ on Sp(B). Hence (g ∗ ϕ)(B) − (Pn ∗ ϕ)(B) → 0. Since ∞ ∞ (Pn ∗ ϕ)(B) = Pn (B − r1H )ϕ(r)dr → g(B − r1H )ϕ(r)dr, as n → ∞, −∞

−∞

with respect to the operator norm, we obtain that ∞ g(B − r1H )ϕ(r)dr, for B ∈ Ssa (β). (g ∗ ϕ)(B) =

(6.2)

−∞

Proposition 6.1. Let g be a Frechet J-Differentiable function on Γ, let J be J0φ , J φ or S and let α ⊂ Γ. Then there exists ρ > 0 such that, for any continuous function ϕ with supp(ϕ) ⊆ [−ρ, ρ], the function g ∗ ϕ is Frechet J-differentiable on Ssa (α) along Jsa and, for each A ∈ Ssa (α) and X ∈ Jsa , ∞ ∇ ∇ gA−r1 (X)ϕ(r)dr. (6.3) (g ∗ ϕ)A (X) = H −∞

Proof. Choose ρ > 0 and compact sets β, γ in Γ such that α ⊂ β ⊂ γ and that Sp(A + Z) ⊆ β and Sp(B + Z) ⊆ γ, for all A ∈ Ssa (α), B ∈ Ssa (β) and Z ∈ Ssa with Z ≤ ρ. It follows from Proposition 4.4 that g is J-Lipschitzian on Γ. For any B ∈ Ssa (β) and X ∈ Jsa , g(B + tX) − g(B) . t By (3.2), there is D > 0 such that |g(B + tX) − g(B)|J ≤ D|tX|J , so that ∇ gB (X) = lim

t→0

∇ gB J ≤ D, for B ∈ Ssa (β).

(6.4)

Let ϕ be a continuous function on R with supp(ϕ) ⊆ [−ρ, ρ] and let A ∈ ∇ ≤ D. Then Ssa (α). Since A − r1H ⊆ Ssa (β), for r ∈ [−ρ, ρ], we have gA−r1 H J ∞ ∇ gA−r1 (X)ϕ(r)dr, for X ∈ Jsa , (6.5) LA (X) = H −∞

is a bounded linear operator from Jsa into J. By Lemma 5.1, for any > 0, there exists δ > 0, δ ≤ ρ, such that, for all R ∈ Ssa (γ) and all X ∈ Jsa with |X|J ≤ δ, ∇ |g(R + X) − g(R) − gR (X)|J ≤ |X|J .

Therefore, for all r ∈ [−ρ, ρ], ∇ |g(A − r1H + X) − g(A − r1H ) − gA−r1 (X)|J ≤ |X|J . H ∞ Set K = −∞ |ϕ(r)|dr. Combining (6.2), (6.5) and (6.6), yields

=|

∞

−∞

(6.6)

|(g ∗ ϕ)(A + X) − (g ∗ ϕ)(A) − LA (X)|J ∞ ∞ ∇ g(A−r1H +X)ϕ(r)dr− g(A−r1H )ϕ(r)dr− gA−r1 (X)ϕ(r)dr|J H −∞

−∞

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∞

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∇ |g(A − r1H + X) − g(A − r1H ) − gA−r1 (X)|J |ϕ(r)|dr ≤ K|X|J , H

for X ∈ Jsa with |X|J ≤ δ. Hence g ∗ ϕ is Frechet J-differentiable at A along Jsa and (6.3) holds. The next result is an analogue of Proposition 6.1 for Operator Lipschitz functions. Its proof is similar to the proof of Proposition 6.1 but uses Proposition 5.4 instead of Lemma 5.1. Proposition 6.2. Let g be a continuously differentiable, Operator Lipschitz function on Γ and let α ⊂ Γ. There exists ρ > 0 such that, for any continuous function ϕ with supp(ϕ) ⊆ [−ρ, ρ], g ∗ ϕ is a continuously differentiable, Operator Lipschitz function on α and ∞ ∇ ∇ gA−r1 (X)ϕ(r)dr, for A ∈ Ssa (α) and X ∈ EA Ssa . (g ∗ ϕ)A (X) = H −∞

Corollary 6.3. Let α be a compact in Γ and let ϕn be non-negative continuous functions on R such that supp(ϕn ) contract to 0 and ∞ ϕn (t)dt = 1. −∞

(i) If g is an Operator Differentiable function on Γ, then sup A∈Ssa (α)

∇ (g ∗ ϕn )∇ A − gA → 0, as n → ∞.

(ii) If g is a continuously differentiable, Operator Lipschitz function on Γ then, for each A ∈ Ssa (Γ) and X ∈ EA Ssa , ∇ (g ∗ ϕn )∇ A (X) − gA (X) → 0, as n → ∞.

Proof. By Proposition 6.1, there exists ρ > 0 such that (6.3) holds for any continuous function ϕ with supp(ϕ) ⊆ [−ρ, ρ], for each A ∈ Ssa (α) and each X ∈ Ssa . Choose N such that supp(ϕn ) ⊆ [−ρ, ρ] for all n ≥ N . Then ∞ ∞ ∇ ∇ ∇ ∇ (g ∗ ϕn )A (X) − gA (X) = gA−s1H (X)ϕn (s)ds − gA (X)ϕn (s)ds −∞

≤

∞

−∞

∇ ∇ gA−s1 (X) − gA (X)ϕn (s)ds ≤ H

−∞

sup s∈supp(ϕn )

∇ ∇ gA−s1 (X) − gA (X). H

Combining this inequality with Theorem 5.2 and taking into account that supp(ϕn ) contract to 0, we prove part (i). Similarly, making use of Propositions 5.4(ii) and 6.2, we prove part (ii).

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7. Algebras of J-Lipschitz and J-differentiable functions Let J be a separable s. n. ideal or J φ , or S and let α be a compact in R. We denote by OLJ (α) the linear space of all J-Lipschitz functions on α; if J = S, we write OL(α). For g ∈ OLJ (α), we denote by kJ (g, α) the minimal constant for which (3.2) holds (if J = S, we denote it by k(g, α)) and set pα (g) = gα + kJ (g, α), where gα = sup |g(t)|. t∈α

Clearly, g(A) ≤ gα for A ∈ Ssa (α). Since kJ (g + h, α) ≤ kJ (g, α) + kJ (h, α), for g, h ∈ OLJ (α), pα is a norm on OLJ (α). If A, B ∈ Ssa (α) and A − B ∈ J, then |g(A)h(A) − g(B)h(B)|J ≤ g(A)|h(A) − h(B)|J + h(B)|g(A) − g(B)|J ≤ (gα kJ (h, α) + hα kJ (g, α))|A − B|J . Therefore kJ (gh, α) ≤ gα kJ (h, α) + hα kJ (g, α). From this it follows that (OLJ (α), pα ) is a commutative normed *-algebra. To prove that OLJ (α) is a Banach algebra, we need the following general result about uniform convergence of sequences of maps (we omit the proof). Lemma 7.1. Let B be a Banach space. (i) Let (Z, ρ) be a quasimetric space (ρ(z, y) may be ∞ for some z, y ∈ Z). The space U (Z, B) of all uniformly continuous, bounded maps F from Z into B with norm F = supz∈Z F (x)B is a Banach space. (ii) Let X be a subset of a normed space Y. For r > 0, let Vr (X, B) be the set of all bounded maps F from X into B such that F (x) − F (y)B ≤ rx − yY , f or x, y ∈ X, supplied with the quasimetric ρ(F, G) =

F (x) − G(x)B . xY x∈X,x=0 sup

Then the space Vr (X, B) is complete. Proposition 7.2. OLJ (α) is a commutative Banach ∗ -algebra. Proof. We only need to prove that OLJ (α) is a complete space. Let {gm } be a fundamental sequence in OLJ (α). Then it is also fundamental in the norm ·α , so there is a continuous function g on α such that g − gm α → 0. For > 0, there is N such that kJ (gn − gm , α) ≤ for n, m ≥ N . Hence kJ (gn , α) = kJ (gN + gn − gN , α) ≤ kJ (gN , α) + kJ (gn − gN , α) ≤ kJ (gN , α) + . Thus there exists r > 0 such that kJ (gm , α) ≤ r for all m. We have that all maps from Jsa (α) into J defined by gm are bounded and |gm (A) − gm (B)|J ≤ r|A − B|J for A, B ∈ Jsa (α). Using the notations of Proposition 7.1 and setting Y = B = J and X = Jsa (α), we obtain that all gm belong to Vr (Jsa (α), J).

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Moreover, for n, m ≥ N and all A ∈ Jsa (α), |gn (A) − gm (A)|J = |(gn (A) − gm (A)) − (gn (0) − gm (0))|J ≤ kJ (gn − gm , α)|A − 0|J ≤ |A|J , so that {gn } is a fundamental sequence in Vr (Jsa (α), J). Hence, by Proposition 7.1(ii), there is a map F from Jsa (α) into J such that |F (A) − F (B)|J ≤ r|A − B|J and |F (A) − gn (A)|J ≤ |A|J , for all A, B ∈ Jsa (α) and for all sufficiently large n. Since g − gn α → 0, it follows that g(A) − gn (A) → 0, for A ∈ Jsa (α). Since also F (A) − gn (A) ≤ |F (A) − gn (A)|J → 0, we have F (A) = g(A). Thus g ∈ OLJ (α). Let P be a rank one projection. If g ∈ OLJ (α) and λ, µ ∈ α, then |g(µ) − g(λ)| = |g(µP ) − g(λP )|J ≤ kJ (g, α)|µ − λ||P |J = kJ (g, α)|µ − λ|. Hence if g is differentiable at a point t ∈ α, then |g (t)| ≤ kJ (g, α).

(7.1)

Theorem 7.3. Let a compact α have a non-empty interior. If J is J0φ , J φ or S, then the Banach ∗ -algebra OLJ (α) is not separable. Proof. Without loss of generality we may assume that α = [−1, 1]. It was proved in [19] that the function 1 g(t) = t2 sin( ), for t = 0, and g(0) = 0 t is S-Lipschitzian on R. Hence it follows from Corollary 6.8(ii) of [19] that g is J-Lipschitzian on R, for any ideal J = J0φ or J φ . For each > 0, set g (t) = g(t + ) − g(). Then g (0) = 0 and, by (3.2), all g are J-Lipschitz functions on R. Hence their restrictions to [-1,1], also denoted by g , belong to OLJ (α). Let 0 < , δ < 18 . It follows from (7.1) that sup |g (t) − gδ (t)|.

pα (g − gδ ) = g − gδ α + kJ (g − gδ , α) ≥

t∈[−1,1]

Set K(, δ) = [−1, 1]\{−, −δ}, ρ = − δ and tn =

1 nπ

− δ. Then

pα (g − gδ ) ≥ supt∈K(,δ) |g (t) − gδ (t)| 1 1 1 1 − cos − 2(t + δ) sin + cos | = sup |2(t + ) sin t+ t+ t+δ t+δ t∈K(,δ) 1 −1 1 1 −1 ) sin(ρ + ) − cos(ρ + ) + (−1)n | nπ nπ nπ n 1 ≥ 1 − 2|ρ| − | cos( )|. ρ

≥ sup |2(ρ +

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Since gδ (−δ) = g (0) = 0, we also have pα (g − gδ ) ≥ |g (−δ) − gδ (−δ)| = |g (−δ)| 1 1 1 1 1 − cos | = |2ρ sin − cos | ≥ | cos | − 2|ρ|. −δ −δ ρ ρ ρ Adding the above inequalities, we have 2pα (g − gδ ) ≥ 1 − 4 | − δ| ≥ 1/2, if , δ ∈ (0, 1/8). Thus there is a continuum of elements in OLJ (α) with the distance between any two of them greater than 14 . It follows from this in a standard way that OLJ (α) is not separable. = |2( − δ) sin

We denote by OLJ (Γ) the commutative algebra of all J-Lipschitz functions on Γ endowed with the family of seminorms pα , where α are compacts in Γ. It follows from Proposition 7.2 that OLJ (Γ) is a complete space. φ φ Lemma 7.4. Let a s. n. ideal J be J0 , J or S. Let α1 , . . . , αn be disjoint compacts in Γ and let α = α1 . . . αn . Then there is C > 0 such that

pα (g) ≤ C(pα1 (g) + . . . + pαn (g)) for g ∈ OLJ (Γ). Proof. Let [A, X] = AX − XA for A, X ∈ B(H). For any compact α in Γ and any g ∈ OLJ (Γ), set mJ (g, α) = inf{D : |[g(A), X]|J ≤ D|[A, X]|J , for X ∈ J and A ∈ Ssa (α)}. It follows from Proposition 4.5 and Theorem 7.5 of [19] that mJ (g, α) ≤ kJ (g, α) ≤ 4mJ (g, α). (7.2) It suffices to prove the lemma for n = 2, so let α = α1 α2 and A ∈ Ssa (α). Then H = H1 ⊕ H2 , A = A1 ⊕ A2 , where Sp(Ai ) ⊆ αi . Any operator can be written in the operator matrix form X = (Xij ), 1 ≤ i, j ≤ 2, so that [A, X]ij = Ai Xij − Xij Aj [g(A), X]i,j = g(Ai )Xij − Xij g(Aj ). Let X ∈ J. Then, for i = 1,2, |[g(Ai ), Xii ]|J ≤ mJ (g, αi )|[Ai , Xii ]|J ≤ mJ (g, αi )|[A, X]|J . Since α1 and α2 are disjoint, it follows from an extension of the Rosenblum’s Theorem (see Lemma 1.3 of [19]) that there is a constant C1 > 0, which only depends on α1 and α2 , such that |Y |J ≤ C1 |T Y − Y R|J , for any Y ∈ J and any T, R such that Sp(T ) ⊆ α1 , Sp(R) ⊆ α2 . Therefore |X12 |J ≤ C1 |A1 X12 − X12 A2 |J = C1 |Q1 [A, X]Q2 |J ≤ C1 |[A, X]|J , where Qi are the projections on Hi . Hence we obtain that |g(A1 )X12 − X12 g(A2 )|J ≤ (g(A1 ) + g(A2 ))|X12 |J ≤ (gα1 + gα2 )C1 |[A, X]|J .

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Similarly, there exists C2 > 0 such that |g(A2 )X21 − X21 g(A1 )|J ≤ (gα1 + gα2 )C2 |[A, X]|J . Therefore |[g(A), X]|J ≤

2

|Qi [g(A), X]Qj |J ≤

i,j=1

2

(mJ (g, αi )+(gα1 +gα2 )Ci )|[A, X]|J .

i=1

Thus mJ (g, α) ≤

2

(mJ (g, αi ) + (gα1 + gα2 )Ci ).

i=1

Set C = max{4, 4C1 + 4C2 + 1}. Taking (7.2) into account, we obtain pα (g) ≤ gα + 4mJ (g, α) ≤ gα + 4

2

(mJ (g, αi ) + (gα1 + gα2 )Ci )

i=1

≤ gα +

2 (4kJ (g, αi ) + (gα1 + gα2 )4Ci ) ≤ C(pJ (g, α1 ) + pJ (g, α2 )). i=1

Let J be J0φ , J φ or S and let F DJ (Γ) and GDJ (Γ) be the spaces of all Frechet and Gateaux J-Differentiable functions on Γ respectively. By Proposition 4.4, F DJ (Γ) ⊆ GDJ (Γ) ⊆ OLJ (Γ). We will show that F DJ (Γ) and GDJ (Γ) are closed subalgebras of OLJ (Γ). Lemma 7.5. The spaces F DJ (Γ) and GDJ (Γ) are subalgebras of OLJ (Γ). Proof. Let gi ∈ F DJ (Γ), i = 1, 2, and let A ∈ Ssa (Γ). For any > 0, there exists δ > 0 such that, for each X in Jsa with |X|J ≤ δ, gi (A + X) − gi (A) ∈ J and |gi (A + X) − gi (A) − (gi )∇ A (X)|J ≤ |X|J . Then g1 (A + X)g2 (A + X) − g1 (A)g2 (A) = g1 (A + X)(g2 (A + X) − g2 (A)) + (g1 (A + X) − g1 (A))g2 (A) ∈ J and ∇ |g1 (A + X)g2 (A + X) − g1 (A)g2 (A) − [(g1 )∇ A (X)g2 (A) + g1 (A)(g2 )A (X)]|J ≤ g1 (A + X)|g2 (A + X) − g2 (A) − (g2 )∇ A (X)|J +|g1 (A + X) − g1 (A) − (g1 )∇ A (X)|J g2 (A) +|g1 (A + X) − g1 (A)|J (g2 )∇ A J |X|J ≤ (g1 (A + X) + g1 (A))|X|J + |g1 (A + X) − g1 (A)|J (g2 )∇ A J |X|J . Since g1 is a J-Lipschitz function, |g1 (A + X) − g1 (A)|J ≤ D|X|J for some D > 0. Thus g1 g2 ∈ F DJ (Γ) and F DJ (Γ) is a subalgebra of OLJ (Γ). Similarly, one can prove that GDJ (Γ) is a subalgebra of OLJ (Γ).

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Let J be J0φ , J φ or S. For α ⊂ Γ and a function g ∈ GDJ (Γ), set dJ (g, α) =

sup A∈Ssa (α)

∇ gA J and λα (g) = gα + dJ (g, α).

(7.3)

By Proposition 4.4, g is J-Lipschitzian on Γ and we obtain from (6.4) that dJ (g, α) ≤ kJ (g, β) < ∞

(7.4)

for any compact β in Γ such that α ⊂ int(β). Therefore λα (g) ≤ pβ (g) < ∞. Proposition 7.6. Let α and β be compacts in Γ such that α ⊂ int(β) and let J be J0φ , J φ or S. Then there exists K > 0 such that, for any g ∈ GDJ (Γ), pα (g) ≤ Kλβ (g).

(7.5)

Proof. There are disjoint segments βi = [ai , bi ], 1 ≤ i ≤ n for some n n, such that n β ⊆ β. Hence it suffices to prove (7.5) for β = α ⊂ i=1 i i=1 βi . The sets αi = α βi are disjoint, so by Lemma 7.4, there is C > 0 such that pα (g) ≤ C(pα1 (g) + . . . + pαn (g)) ≤ C(pβ1 (g) + . . . + pβn (g)). Let A, B ∈ Ssa (βi ) and A − B ∈ J. The operators G(s) = sA + (1 − s)B belong to Ssa (βi ) for s ∈ [0, 1]. Since g is Gateaux J-differentiable on Γ, it follows from the law of the mean (see [20]) that ∇ |A − B|J ≤ dJ (g, βi )|A − B|J . |g(A) − g(B)|J ≤ sup gG(s) s∈[0,1]

Therefore kJ (g, βi ) ≤ dJ (g, βi ), so pβi (g) ≤ λβi (g) ≤ λβ (g), for 1 ≤ i ≤ n. Thus pα (g) ≤ Cnλβ (g). Corollary 7.7. Let J be J0φ , J φ or S. The topology in GDJ (Γ) induced by the seminorms pα is equivalent to the topology defined by the seminorms λα . Let J be a s. n. ideal or S and let B(J) be the Banach algebra of all bounded operators on J. If J = S and A ∈ J, we set |A|J = ∞. We denote by UJ (Γ) the linear space of all maps Φ from Ssa (Γ) into B(J) which are 1) “bounded on compacts”: for any compact α in Γ, qα (Φ) =

sup A∈Ssa (α)

Φ(A)J < ∞;

(7.6)

2) “uniformly continuous on compacts”: for any > 0, there is δ = δ(, α) such that A, B ∈ Ssa (α) and |A − B|J < δ imply Φ(A) − Φ(B)J < .

(7.7)

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The space UJ (Γ) is complete in the topology defined by the seminorms qα . Indeed, for α ⊂ Γ, Ssa (α) is a metric space with ρ(A, B) = |A − B|J . By Lemma 7.1(i), the space of all bounded, uniformly continuous maps from Ssa (α) into B(J) is a Banach space. Let Φλ be a fundamental net of maps in UJ (Γ). The set of their restrictions to Ssa (α) is a fundamental net with respect to qα and, therefore, converges to a bounded, “uniformly continuous” map Φα from Ssa (α) into B(J). Thus Φλ converge to a map Φ in UJ (Γ), so that UJ (Γ) is complete. Let J be J0φ , J φ or S and let g ∈ F DJ (Γ). The G-derivative g ∇ maps Ssa (Γ) ∇ into B(J) : A ∈ Ssa (Γ) → gA . Comparing (7.3) and (7.6), we have that, for any ∇ α in Γ, qα (g ) = dJ (g, α) < ∞, so that the map g ∇ is “bounded on compacts”. It follows from Theorem 5.2 that g ∇ is also “uniformly continuous on compacts”. Hence g ∇ ∈ UJ (Γ). Lemma 7.8. Let J be J0φ , J φ or S. A function g on Γ belongs to F DJ (Γ) if and only if there exists a map Φ ∈ UJ (Γ) such that 1 g(A + X) − g(A) = Φ(A + tX)(X)dt, (7.8) 0

for each A ∈ Ssa (Γ) and each X ∈ Jsa such that all A + tX ∈ Ssa (Γ), t ∈ [0, 1]. ∇ Proof. If g is Frechet J-Differentiable on Γ, then the map g ∇ : A → gA belongs to UJ (Γ) and (7.8) follows from the general Newton-Leibniz formula for maps with continuous G-derivative. The converse statement can be proved in the same way as the converse statement in Corollary 5.3.

Theorem 7.9. Let J be J0φ , J φ or S. The algebra F DJ (Γ) is closed in OLJ (Γ) and, therefore, complete. Proof. For any α ⊂ Γ, there is a compact β in Γ such that α ⊂ int(β). It follows from (7.3), (7.4) and (7.6) that, for each g ∈ F DJ (Γ), qα (g ∇ ) = dJ (g, α) ≤ kJ (g, β) ≤ pβ (g).

(7.9)

Let a net of functions {gλ } from F DJ (Γ) converge to a function g in OLJ (Γ). By (7.9), {(gλ )∇ } is a fundamental net of maps in UJ (Γ). Since UJ (Γ) is complete, the net converges to a map Φ ∈ UJ (Γ). We have from Lemma 7.8 that 1 gλ (A + X) − gλ (A) = (gλ )∇ A+tX (X)dt, 0

for each A ∈ Ssa (Γ) and X ∈ Jsa such that A + tX ∈ Ssa (Γ), for all t ∈ [0, 1]. Fix A and X and let a compact α in Γ contain all Sp(A + tX), for t ∈ [0, 1]. Since gλ uniformly converge to g on α and (gλ )∇ converge to Φ, we have that, for any > 0, there exists λ() such that g(B)−gλ (B) ≤ and Φ(B)−(gλ )∇ B J ≤ ,

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for all B ∈ Ssa (α) and all λ ≥ λ(). Since T ≤ |T |J , for T ∈ J, we have 1 g(A + X) − g(A) − 0 Φ(A + tX)(X)dt ≤ g(A + X) − gλ (A + X) + gλ (A) − g(A) 1 Φ(A + tX)(X) − (gλ )∇ + A+tX (X)dt 0

≤ 2 +

0

≤ 2 +

0

1

1

|Φ(A + tX)(X) − (gλ )∇ A+tX (X)|J dt Φ(A + tX) − (gλ )∇ A+tX J |X|J dt ≤ 2 + |X|J .

Thus g(A + X) − g(A) =

1 0

Φ(A + tX)(X)dt and, by Lemma 7.8, g ∈ F DJ (Γ).

Corollary 7.10. Let J be J0φ , J φ or S. Any function g on Γ with continuous second derivative is Frechet J-Differentiable and is a limit in F DJ (Γ) of a sequence of polynomials. Proof. The first statement of the corollary was proved (at least for J = S) by Daletskii and Krein [8]. We will prove both statements simultaneously. Let Pn be polynomials which uniformly converge to g on all compacts in Γ together with the first and the second derivatives. For α ⊂ Γ, choose a compact β in Γ such that α ⊂ int(β). We have (see (2.8)) that g − Pn 2,β → 0. Therefore it follows from (2.9) that dJ (g − Pn , α) =

sup A∈Ssa (α)

(g − Pn )∇ A J ≤ C(α, β)g − Pn 2,β → 0.

Hence λα (g − Pn ) → 0. Since all Pn ∈ F DJ (Γ), we obtain from Corollary 7.7 and Theorem 7.9 that g belongs to F DJ (Γ). Theorem 7.11. Let J = S 1 be a separable ideal, or J = S. The algebra GDJ (Γ) is closed in OLJ (Γ). If J is separable, then GDJ (Γ) is non-separable. Proof. If J = S, then GDJ (Γ) = F DJ (Γ) and the proof follows from Theorem 7.9. Let J = S 1 be separable. By Theorem 3.6, a function from OLJ (Γ) is Gateaux JDifferentiable if and only if it is differentiable. Let gλ belong to GDJ (Γ) and gλ → g in OLJ (Γ). Let t0 ∈ Γ. Since Γ is open, there is a segment α = [a, b] ⊂ Γ such that t0 ∈ (a, b). Let a subsequence {gn } converges to g in pα . Then gn uniformly converge to g, all gn are differentiable on Γ and, by (7.1), (t)| ≤ kJ (gn − gm , α) → 0. sup |gn (t) − gm t∈α

gn

Hence converge uniformly on α. The limit g of functions gn on α with uniformly converging (not necessarily continuous) derivatives is differentiable and g (t) = lim gn (t), for t ∈ α (see, for example, [12, Theorem XII, 2,8*]). Thus g is differentiable on Γ, so that g ∈ GDJ (Γ). Hence GDJ (Γ) is closed in OLJ (Γ).

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To prove that GDJ (Γ) is non-separable, it suffices to note that all functions g , constructed in the proof of Theorem 7.3, are differentiable and J-Lipschitzian, so they are Gateaux J-Differentiable. The spaces of Frechet and Gateaux S-Differentiable functions on Γ coincide. We call these functions Operator Differentiable and denote the space of all such functions by OD(Γ). Finally, we show that OD(Γ) is separable. Theorem 7.12. Polynomials are dense in OD(Γ). Proof. Let E be the space of all functions in OD(Γ) with compact support in Γ and let E2 be the subspace of E of all functions with continuous second derivative. By Corollary 7.10, polynomials are dense in E2 . Let us show that E2 is dense in E. For g ∈ E, let α = supp(g). By Corollary 6.3(i), there are non-negative functions ϕn on R such that supp(ϕn ) contract to 0, ∞ ∇ ϕn (t)dt = 1 and dS (g ∗ ϕn − g, α) = sup (g ∗ ϕn )∇ A − gA → 0. A∈Ssa (α)

−∞

One can choose ϕn to be infinitely differentiable. Hence g ∗ ϕn ∈ E2 and λα (g ∗ ϕn − g) = g ∗ ϕn − gα + dS (g ∗ ϕn − g, α) → 0. Thus E2 is dense in E. Finally, to prove that E is dense in OD(Γ), observe that, for any α ⊂ Γ, there exists ψ ∈ E with ψ(t) = 1 in a neighbourhood of α. By Lemma 7.5, the function gψ belongs to E, for any g ∈ OD(Γ). It is easy to see that λα (g − gψ) = 0. Since α is arbitrary, g belongs to the closure of E. In the case when dim H < ∞, all spaces OLJ (Γ) coincide with the space Lip(Γ) of all functions Lipschitzian in the usual sense on all compact subsets of Γ (see [19, Corollary 4.6]). By Corollary 5.7, all spaces GDJ (Γ) and F DJ (Γ) in this case coincide with the space D1 (Γ) of all differentiable functions on Γ, so that the analogue of Theorem 7.12 does not hold. The analogue ofTheorem 7.11 remains valid if OD(Γ) = D1 (Γ) is replaced by the algebra D1 (Γ) Lip(Γ). We proved that “most” Operator Lipschitz functions on Γ are not Operator Differentiable, since Operator Differentiable functions form a separable closed subspace in the non-separable space of Operator Lipschitz functions. Problem 7.13. (i)Is the algebra F DJ (Γ) separable, for a separable J? (ii) Does R act continuously by shifts on F DJ (Γ) for a separable J : ∇ ∇ lim gA−r1 − gA J = 0, if g ∈ F DJ (Γ) and A ∈ Ssa (Γ)?

r→0

The above problems are linked. If we have a positive answer to the problem (ii), we will be able to prove an analogue of Corollary 6.3(i) for a separable J which, in turn, will allow us to prove (as we did in Theorem 7.12) that the space E2 of all functions with continuous second derivative is dense in the space of all functions in F DJ (Γ) with compact support.

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˜ If, instead of F DJ (Γ), we consider the algebra of all Frechet J-Differentiable functions, where J˜ = (C1)+J, then the answer to Problems 7.13 will be positive. In ∇ ∇ −gA J˜ = 0, this case, repeating the proof of Theorem 5.2, we obtain limr→0 gA−r1 for g ∈ F DJ˜(Γ) and A ∈ Ssa (Γ). Hence the fact that the polynomials are dense in F DJ˜(Γ) can be easily obtained by repeating the proof of Theorem 7.12 with ˜ Corollary 6.3(i) replaced by its analogue for J. ˜ Problem 7.14. Are Frechet J-Differentiable functions J-Differentiable?

8. Action of operator Lipschitz and operator differentiable functions in the domains of derivations of C ∗ -algebras Arazy, Barton and Friedman [2] proved that Operator Differentiable functions act on the domains of the generators of all strongly continuous one-parameter semigroups of *-endomorphisms of C ∗ -algebras. We extend their result and establish that these functions act on the domains of all closed *-derivations of C ∗ -algebras. We also show that a function acts on the domains of all weakly closed *-derivations if and only if it is Operator Lipschitzian. Let U be a C ∗ -subalgebra of B(H). A map δ from a dense *-subalgebra D(δ) of U (called the domain of δ) into B(H) is a closed *-derivation if δ(AB) = Aδ(B) + δ(A)B, δ(A∗ ) = δ(A)∗ , for A, B ∈ D(δ), and its graph G(δ) = {(A, δ(A)) : A ∈ D(δ)} is norm closed in B(H) ⊕ B(H). It is weakly closed if G(δ) is closed in B(H) ⊕ B(H) in the weak operator topology. The domains of closed *-derivations constitute one of the main classes of Differentiable Banach algebras (see [5, 16]). The domains of weakly closed derivations are called W ∗ domain algebras by Weaver [28]. In the theory of “non-commutative metric spaces” they play the same role as Lipschitz algebras in the theory of metric spaces (see [7, 28]). Recall that a function g on Γ acts on the domain D(δ) of δ, if g(A) ∈ D(δ), for any A = A∗ ∈ D(δ) such that Sp(A) ⊂ Γ. Theorem 8.1. Any Operator Differentiable function g on an open set Γ acts on the domains D(δ) of all closed ∗ -derivations δ of C ∗ -algebras and ∇ (δ(A)), f or A = A∗ ∈ D(δ) with Sp(A) ⊂ Γ. δ(g(A)) = gA

(8.1)

Proof. Let g ∈ OD(Γ) and let polynomials Pn converge to g in OD(Γ) (see Theorem 7.12): λα (g − Pn ) = g − Pn α + dS (g − Pn , α) = g − Pn α +

sup A∈Ssa (α)

∇ gA − (Pn )∇ A → 0

for every α ⊂ Γ. It is easy to check that (8.1) holds for all polynomials and all derivations: δ(Pn (A)) = (Pn )∇ A (δ(A)). Hence, for each A ∈ Ssa (α), we have ∇ Pn (A) → g(A) and δ(Pn (A)) = (Pn )∇ A (δ(A)) → gA (δ(A))

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∇ in the operator norm. Since δ is closed, g(A) ∈ D(δ) and δ(g(A)) = gA (δ(A)).

In the rest of the section we study the action of Operator Lipschitz functions on the domains of derivations. Proposition 8.2. Let g be a continuously differentiable, Operator Lipschitz function ∗ ∗ on Γ, let δ be a closed ∗ -derivation on an operator C -algebra and A = A ∈ D(δ) with Sp(A) ⊂ Γ. If δ(A) ∈ EA Ssa (see (3.10)) then g(A) ∈ D(δ) and ∇ δ(g(A)) = gA (δ(A)).

Proof. Choose a compact α in Γ whose interior contains Sp(A). Without loss of generality we may assume that supp(g) = α. Let (ϕn )∞ n=1 be non-negative, infinitely differentiable functions on R such that supp(ϕn ) contract to 0 and ∞ ϕn (t)dt = 1. −∞

Then the convolutions gn = g ∗ ϕn are infinitely differentiable functions with compact support, uniformly converging to g on α. By Theorem 8.1, all functions gn act on D(δ), sothat gn (A) ∈ D(δ) and δ(gn (A)) = (gn )∇ A (δ(A)). If δ(A) ∈ EA Ssa , it follows from Corollary 5.7(ii) that (gn )∇ A (δ(A)) converge ∇ to gA (δ(A)). Thus ∇ gn (A) → g(A) and δ(gn (A)) = (gn )∇ A (δ(A)) → gA (δ(A)) ∇ (δ(A)). in the operator norm. Since δ is closed, g(A) ∈ D(δ) and δ(g(A)) = gA

A ∗ -derivation δ is locally inner if δ(A) ∈ EA Ssa for every A = A∗ ∈ D(δ). All derivations of commutative C ∗ -algebras into themselves are locally inner, as well as all approximately inner derivations (δ is approximately inner, if there are (Bn ) in U such that i[Bn , A] → δ(A), for every A ∈ D(δ)). Any derivation δ which maps D(δ) into C(H)(= S ∞ ) is locally inner, since, by (3.6), C(H) = ((A) C(H)) + δA (C(H)) ⊆ EA C(H). Corollary 8.3. Any continuously differentiable, Operator Lipschitz function on Γ acts on the domains of all locally inner derivations. For any selfadjoint (unbounded) operator T on H, set D(δT ) = {A ∈ B(H) : AD(T ) ⊆ D(T ), and (T A − AT )|D(T ) is a bounded operator}. Then D(δT ) is a ∗-subalgebra of B(H) and the map δT defined by δT (A) = i(T A − AT ), for A ∈ D(δT ),

(8.2)

is a closed *-derivation from the C ∗ -algebra UT = D(δT ) into B(H) with domain D(δT ) (see [18]). Moreover, δT is weakly closed. Indeed, let (R, S) belong to the

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closure of the graph {(A, δT (A)) : A ∈ D(δT )} of δT in the weak operator topology. Then, for any x, y ∈ D(T ), there are An in D(δT ) such that (An x, T y) → (Rx, T y), (An T x, y) → (RT x, y), (δT (An )x, y) → (Sx, y). Taking into account (8.2) and the fact that T is selfadjoint, we obtain (Rx, T y) = lim(An x, T y) = lim(T An x, y) = lim(−iδT (An )x, y) + lim(An T x, y) = (−iSx, y) + (RSx, y). Therefore Rx ∈ D(T ) and Sx = i(T R − RT )x, so R ∈ D(δT ) and S = δT (R). Thus δT is weakly closed. Theorem 8.4. A continuous function g on Γ acts on the domains of all weakly closed ∗ -derivations of C ∗ -algebras if and only if it is Operator Lipschitzian on Γ. Proof. Suppose that g acts on the domains of all weakly closed *-derivations of C ∗ -algebras. If g is not Operator Lipschitzian on Γ, then there is a compact α in Γ, where g is not Operator Lipschitzian. By (3.3), there are An ∈ Ssa (α) and Xn ∈ B(H)sa such that [g(An ), Xn ] ≥ n[An , Xn ]. Without loss of generality, we may assume that [An , Xn ] = 1 for all n, so that (8.3) [g(An ), Xn ] ≥ n. ∞ ∞ Set d = maxt∈α |t| , A = n=1 An , X = n=1 Xn and H = n=1 Hn , where all Hn = H. Then An ≤ d,so that A is a bounded selfadjoint operator on H ∞ with Sp(A) ⊆ α and g(A) = n=1 g(An ) ∈ B(H). The operator X is selfadjoint, possibly unbounded and ∞

D(X) = {ξ =

∞

ξn : ξn ∈ H,

n=1

∞

ξn 2 < ∞, Xξ2 =

n=1

∞

Xn ξn 2 < ∞}.

n=1

Since Xn An = An Xn + [Xn , An ] and since [An , Xn ] = 1, it follows that, for ξ ∈ D(X), XAξ2 =

∞

Xn An ξn 2 ≤ 2

n=1

≤ 2d2

∞ n=1

∞

An Xn ξn 2 + 2

n=1 ∞

Xn ξn 2 + 2

∞

[Xn , An ]ξn 2

n=1

ξn 2 < ∞.

n=1

∞ Hence AD(X) ⊆ D(X). We also have that [X, A] = n=1 [Xn , An ] is bounded on D(X). The derivation δX , defined by (8.2), is weakly closed and, by the above argument, A ∈ D(δX ). If g acts on the domains of all weakly closed *-derivations, then g(A) ∈ D(δX ), so that δX (g(A)) ∈ B(H). On the other hand, by (8.3), δX (g(A)) = [X, g(A)] =

∞

[Xn , g(An )]

n=1

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is an unbounded operator. This contradiction proves that g is Operator Lipschitzian. Conversely, let g be Operator Lipschitzian on Γ. Let δ be a weakly closed derivation, A = A∗ ∈ D(δ) and Sp(A) ⊂ Γ. Choose α in Γ such that Sp(A) ⊂ int(α). Choose ρ > 0 and a compact β in Γ such that α ⊂ β and Sp(B − r1H ) ⊆ β, for all r ∈ [−ρ, ρ] and all B ∈ Ssa (α). Without loss of generality we may assume that supp(g) ⊆ β. Let (ϕn )∞ n=1 be non-negative, infinitely differentiable functions on R such that supp(ϕn ) lie in [−ρ, ρ] and contract to 0 and such that ∞ ϕn (t)dt = 1. −∞

Then the convolutions gn = g ∗ ϕn are infinitely differentiable functions with compact support, uniformly converging to g on β. It follows from Theorem 8.1 that the functions gn act on D(δ) : gn (A) ∈ D(δ) and δ(gn (A)) = (gn )∇ A (δ(A)). The functions gn are Operator Lipschitzian on α. Hence, by (7.4), ∇ δ(gn (A)) = (gn )∇ A (δ(A)) ≤ (gn )A δ(A) ≤ k(gn , α)δ(A),

where k(gn , α) are the minimal constant for which (3.1) holds for gn . Since g is Operator Lipschitzian, we obtain from (3.1) and (7.2) that for B, C ∈ Ssa (α) ∞ ∞ gn (B) − gn (C) = g(B − r1H )ϕn (r)dr − g(C − r1H )ϕn (r)dr −∞ −∞ ∞ g(B − r1H ) − g(C − r1H )|ϕn (r)|dr ≤ −∞ ∞ k(g, β)B − C|ϕn (r)|dr = k(g, β)B − C. ≤ −∞

Hence k(gn , α) ≤ k(g, β). Therefore δ(gn (A)) ≤ k(g, β)δ(A) for all n. Since the sequence (δ(gn (A))) is bounded, it has a cluster point R in the weak operator topology. We also have that gn (A) → g(A) with respect to the operator norm. Hence the pair (g(A), R) belongs to the closure of the graph of δ in the weak operator topology. Since δ weakly closed, g(A) ∈ D(δ). Let a function g on Γ act on the domains of all closed *-derivations. By Theorem 8.4, it is Operator Lipschitzian. The *-derivation δ : h(t) → h (t) on the C ∗ -algebra of all continuous functions on [a, b] ⊂ Γ with domain D(δ) = C (1) (a, b) is closed. Taking h0 (t) ≡ t, we have g(h0 ) ∈ C (1) (a, b). Hence g is continuously differentiable. Denote by CD(Γ) and W CD(Γ) the space of all functions on Γ which act on the domains of all closed and weakly closed *-derivations of C ∗ -algebras, respectively. From the above argument and from Theorems 8.1 and 8.4 it follows that W CD(Γ) = OL(Γ) and OD(Γ) ⊆ CD(Γ) ⊆ OL(Γ) C (1) (Γ). Problem 8.5. Which of the above inclusions are proper?

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[20] A.N. Kolmogorov and S.V. Fomin, “Elements of the theory of functions and functional analysis”, Nauka, Moscow, 1976. [21] A. McIntosh, Functions and derivations of C ∗ -algebras, J. Funct. Anal., 30 (1978), 264–275. [22] G.K. Pedersen, Operator Differentiable Functions, Publ. RIMS, Kyoto Univ., 36 (2000), 139–157. [23] V.V. Peller, Hankel operators in the perturbation theory of unitary and selfadjoint operators, Funkzionalnii Anal. i ego priloz., 19(2) (1985), 37–51. [24] R.T. Powers, A remark on the domain of an unbounded derivation of a C ∗ -algebra, J. Funct. Anal., 18 (1975), 85–95. [25] F. Riesz and B. Sz.-Nagy, “Lecons d’analyse fonctionnelle”, Akademiai Kiado, Budapest, 1972. [26] S. Sakai, “Operator algebras in dynamical systems”, CUP, Cambridge, New York, 1991. [27] T.V. Shulman, On covariant maps of matrices, Spectral and Evolutionary Problems, Proceedings of the 11-th Crimean Autumn Mathematical School- Symposium, (2001), 77–82. [28] N. Weaver, Lipschitz Algebras and Derivations of von Neumann Algebras, J. Funct. Anal., 139 (1996), 261–300. [29] H. Widom, When are differentiable functions differentiable, In: Linear and Complex Analysis Problem Book, Lect. Notes Math., 1043 (1984), 184–188. E. Kissin School of Communications Technology and Mathematical Sciences University of North London Holloway Road, London N7 8DB Great Britain e-mail: [email protected] V.S. Shulman School of Communications Technology and Mathematical Sciences University of North London Holloway Road, London N7 8DB Great Britain and Department of Mathematics Vologda State Technical University Vologda Russia e-mail: shulman [email protected] Submitted: March 1, 2001 Revised: August 28, 2002