AN EXAMPLE IN THE THEORY OF AC-OPERATORS 1. Introduction ...

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 129, Number 5, Pages 1453–1457 S 0002-9939(00)05733-6 Article electronically published on October 24, 2000

AN EXAMPLE IN THE THEORY OF AC-OPERATORS IAN DOUST AND T. A. GILLESPIE (Communicated by Joseph A. Ball)

Abstract. AC-operators are a generalization in the context of well-boundedness of normal operators on Hilbert space. It was shown by Doust and Walden that compact AC-operators have a representation as a conditionally convergent sum reminiscent of the spectral representations for compact normal operators. In this representation, the eigenvalues must be taken in a particular order to ensure convergence of the sum. Here we show that one cannot replace the ordering given by Doust and Walden by the more natural one suggested in their paper.

1. Introduction There is a long history of results showing that a compact Banach space operator T with a suitable functional calculus has a representation of the form (∗)

T =

∞ X

λj Pj

j=1

where {λj } is the set of non-zero eigenvalues of T , and Pj is the Riesz projection onto the eigenspace corresponding to λj . (Of course, T may have only finitely many non-zero eigenvalues.) For self-adjoint or normal compact operators the sum (∗) converges unconditionally. To provide a theory which covered operators whose spectral expansions may only converge conditionally, Smart [Sm] introduced the concept of a well-bounded operator. Let X denote a complex Banach space. An operator T ∈ B(X) is said to be well-bounded if it admits an absolutely continuous functional calculus, i.e. if there exist K > 0 and a compact interval [a, b] ⊂ R such that ) ( Z b

kg(T )k ≤ K

|g(a)| +

|g 0 (t)| dt

a

≡ K kgkAC[a,b]

Received by the editors August 27, 1999. 2000 Mathematics Subject Classification. Primary 47B40. Key words and phrases. AC-operators, compact operators. The work of the first author was supported by the Australian Research Council. The second author thanks the School of Mathematics, University of New South Wales for its hospitality when this work was undertaken. c

2000 American Mathematical Society

1453

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I. DOUST AND T. A. GILLESPIE

for all polynomials g. One restriction of well-bounded operators is that they necessarily have real spectrum. The analogue of a normal operator in this context is known as an AC-operator. An operator T ∈ B(X) is an AC-operator if T = A+ iB for some pair {A, B} of commuting well-bounded operators, or equivalently, if T has a functional calculus for the absolutely continuous functions on some rectangle [a, b] × [c, d] ⊂ R2 . We refer the reader to [Dow] for background on well-bounded operators and to [BG] for background on AC-operators. It was shown by Cheng and Doust [CD1] that a compact well-bounded operator has a representation in the form (∗) where the sum is ordered so that (O-1)

|λ1 | ≥ |λ2 | ≥ · · · .

Examples show that we may indeed get only conditional convergence for the sum. Doust and Walden [DW] extended this result to cover compact AC-operators. In this case they could only prove convergence under a more complicated ordering of the eigenvalues. For a complex number λ = x+ iy with x, y ∈ R, let |λ|∞ = max {|x| , |y|}. Define the order ≺ on C by setting λ ≺ µ if (i) |λ|∞ < |µ|∞ , or, (ii) if |λ|∞ = |µ|∞ = α and µ lies on that part of the square |z|∞ = α between −α + iα and λ going from −α + iα in a clockwise direction. In [DW] it was shown that if T is a compact AC-operator, then T has a representation in the form (∗) where the eigenvalues are ordered by (O-2)

λ1 λ2 . . . .

They also showed that under certain circumstances the sum also converges to T under the order (O-1), but they left open the question as to whether this is always the case. In this note we construct an example which shows that in general one cannot use the order (O-1) for compact AC-operators. 2. The example The following general result will be used in the construction. We leave the proof to the reader. Lemma. Suppose that {zj }∞ j=1 is a sequence in a Banach space Z and that z = P∞ z converges. Given a sequence 0 = m0 < m1 < m2 < . . . of integers, define j=1 j in Z by setting wmk−1 +` = zmk +1−` for k = 1, 2, . . . and the sequence {wj }∞ j=1 P∞ ` = 1, . . . , mk − mk−1 . Then j=1 wj converges and equals z. Theorem. Suppose that X is an infinite dimensional Banach space with a basis. Then there exists a compact AC-operator T ∈ B(X) for which the sum (∗) above does not converge under the order (O-1). Proof. As X has a basis, it also has a conditional basis {xj } [PS]. Denote Pn the corresponding coordinate projections by {Pj } and for n ≥ 1 set Qn = j=1 Pj . We shall apply the convention that P0 = Q0 = 0. We note that the sets {Pj } and {Qj } are uniformly bounded by say K.


COMPACT AC-OPERATORS

1455

rk−1 • rk + dk i •........ .......

....... ....... ....... ....... ........ ....... ....... ........ ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ...... . .. . . ... ... .. ... .... .... .. .. . ..... .. .... . ... . ... . . . .... . . ... . . .. .... . . . . .. .. ....... ....... ....... ............. ....... ........... ....... ....... .......... ....... ....... ....... ....... ....... ....... ....... ....... ......... ....... ....... .... . . . .... . .. ... . . ... .. ... .... .. mk−1 +1 .. ... ... . . . .. ... . .... .... .... .... ... .. m +2 k−1 . .......... ....... ....... .......... ....... .................. ....... .......... ....... ....... ....... ....... ....... ....... ....... ....... ......... ....... ....... .. ... . . . . ... .. . . . . . . ... ... .... .... .... ... ... .. .... . . . ... ... .. ... ... ... ... . . . .. .. .. . ....... ....... ....... .......... ....... ....... ....... ....... ........... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... ....... . ... .... .. ... .. .. .. . . .... ... ... ... ... mk−1 +3 .... . ... . . . .. .. . . .. . . .... ... ... ... .. ... ... .. .. .... ... . ... ... ... ... ... . . . . . .... .. . .. .. .. .. .. .. ... .. .. .. .. .. .. .. .... ... ... ... ... ... ... . . . . . . . . ......... ....... ....... ......... ....... ......... ....... ....... ......... ....... ....... ....... ....... ....... ......... ....... ....... ......... ....... ....... . ... .. .. .. .. .. ... .. . . .... . ... ... ... ... ... .. .. . . . . . .... . .. .. .. .. m . k ......... ....... ....... ......... ....... ......... ....... ....... ......... ....... ....... ....... ....... ....... ....... ....... ....... ........... ....... ....... .... . ... ... .... ... ... ... ... ... . .. . . . . .. . .. .. .. . . . ... .. ... ... .. .. .. .... ... . .. .. .. .. .. . .... ....... ....... ........ ....... ....... ........ ....... ....... .......... ....... ....... ....... ....... ....... ....... ....... ......... ....... ....... ......

λ

•

•λ

•

λ

•λ

rk •

• rk + dk

Figure 1. As {xj } is conditional

there exists m1 ∈ N and a subset N1 ⊂ {1, 2, . . . , m1 } = I1

P

such that j∈N1 Pj ≥ 2. For k ≥ 2 we can recursively find mk > mk−1 and

P

Nk ⊂ {mk−1 + 1, . . . , mk } = Ik such that j∈Nk Pj ≥ k. For convenience, set m0 = 0. √ For k ≥ 1, let rk = eiπ/4 / k and suppose that dk is a small positive number whose value will be fixed later in the proof. For the present it suffices to require that √ (a) for k ≥ 2, k −1/2 + 2dk < (k − 1)−1/2 , (b) dk < (2 |Nk |)−1 . Let dk . δk = mk − mk−1 + 1 For k ≥ 1 and j = 1, 2, . . . , mk − mk−1 , let √ (−1)κ(k,j) δk i, λmk−1 +j = rk + dk i + j 2e−iπ/4 δk − 3 where κ(k, j) is 1 if mk−1 + j ∈ Nk and 0 otherwise. Figure 1 shows a typical / Nk ). arrangement of these values (with λmk−1 +2 ∈ Nk and λmk−1 +1 , λmk−1 +3 ∈ Note that λ` will be above the diagonal drawn if and only if ` ∈ Nk . It is not too difficult to show that if we choose dk small enough, then |rk + dk i| < |λ` | whenever ` ∈ Nk and hence (1)

|λ`1 | < |λ`2 |

whenever `1 ∈ Ik \ Nk and `2 ∈ Nk .


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I. DOUST AND T. A. GILLESPIE

For j = 1, 2, . . . , let P∞aj = Re(λj ) and bj = Im(λj ). We wish to define A = P∞ a P and B = j=1 j j j=1 bj Pj . As b1 > b2 > . . . , it follows from [CD2] that the second partial sum converges (in norm) and that B is well-bounded. The case of the first sum is just a little more delicate. Let {˜ aj } be the decreasing rearrangement of {aj } and let {P˜j } be the corresponding rearrangement of the coordinate projections. It is easy to check that the partial sum projections corresponding to this ordering are uniformly bounded (by P∞ ˜j P˜j converges to a well-bounded operator as well. It now 3K). Thus A˜ = j=1 a P∞ ˜ and so A is follows from the lemma that A = j=1 aj Pj converges and equals A, a well-bounded operator. Clearly A and B commute and so T = A + iB is a compact AC-operator with eigenvalues {λj }. It remains now to show that if we try to order these eigenvalues by (O-1), then the corresponding sum (∗) does not converge. It follows from (1) that for each k ≥ 1 X X λj Pj + λj Pj Sk = j≤mk−1

j∈Nk

P is a partial sum of (∗) under (O-1). Note that for each k, j≤mk−1 λj Pj is a partial sum of (∗) under (O-2) so these sums are uniformly bounded say by C. By property (b) above, |λj − rk | < 2dk < 1/ |Nk | for j ∈ Ik . Thus

X X X X

λj Pj + rk Pj + λj Pj − rk Pj kSk k = j≤mk−1

j∈Nk

j∈Nk

j≤mk−1

j∈Nk

j∈Nk

X

X X

Pj − λj Pj − (λj − rk )Pj ≥ rk X k |λj − rk | K ≥ √ −C − k j∈Nk √ ≥ k − C − K.

j∈Nk

As this sequence of partial sums is unbounded, the series cannot converge under (O-1). The hypotheses on X could of course be relaxed somewhat. All that is required is that X admits a uniformly bounded increasing sequence of projections with bad unconditionality properties.

References [BG] [CD1] [CD2] [Dow] [DW]

E. Berkson and T.A. Gillespie, Absolutely continuous functions of two variables and wellbounded operators, J. London Math. Soc (2) 30 (1984), 305–321. MR 86c:47044 Cheng Qingping and I. Doust, Well-bounded operators on nonreflexive Banach spaces, Proc. Amer. Math. Soc. 124 (1996), 799–808. MR 96f:47065 Cheng Qingping and I. Doust, Compact well-bounded operators, Preprint. H.R. Dowson, Spectral theory of linear operators, London Mathematical Society Monographs 12, Academic Press, London, 1978. MR 80c:47022 I. Doust and B.L. Walden, Compact AC-operators, Studia Math. 117 (1996), 275–287. MR 97a:47047


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[PS] [Sm]

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A. Pelczy´ nski and I. Singer, On non-equivalent and conditional bases in Banach spaces, Studia Math. 25 (1964), 5–25. MR 31:3831 D.R. Smart, Conditionally convergent spectral expansion, J. Austral. Math. Soc. 1 (1960), 319–333. MR 23:A3462

School of Mathematics, University of New South Wales, Sydney, New South Wales 2052, Australia E-mail address: [email protected] Department of Mathematics and Statistics, University of Edinburgh, James Clerk Maxwell Building, Edinburgh, EH9 3JZ, Scotland E-mail address: [email protected]
