standard models under polynomial positivity conditions - CiteSeerX

c Copyright by Theta, 1999

J. OPERATOR THEORY 41(1999), 365–389

STANDARD MODELS UNDER POLYNOMIAL POSITIVITY CONDITIONS SANDRA POTT

Communicated by Florian-Horia Vasilescu Abstract. We develop standard models for commuting tuples of bounded linear operators on a Hilbert space under certain polynomial positivity conditions, generalizing the work of V. M¨ uller and F.-H. Vasilescu in [6], [14]. As a consequence of the model, we prove a von Neumann-type inequality for such tuples. Up to similarity, we obtain the existence of in a certain sense “unitary” dilations. Keywords: Multivariable spectral theory, weighted multishifts, standard models, dilations, functional calculus. MSC (2000): 47A45, 47A60.

1. INTRODUCTION

Let H be a separable Hilbert space and T = (T1 , . . . , Tn ) a commuting tuple of n P bounded linear operators on H. T is called a spherical contraction, if Ti ∗ Ti 6 1H , and a spherical unitary, if

n P

i=1 ∗

Ti Ti = 1H and in addition, all components

i=1

of T are normal. We say that T has a spherical dilation if there is a spherical unitary U which dilates T , i.e. T α = PH U α |H for all α ∈ Nn0 . There is no easy generalization of the famous Dilation Theorem for contractions of Sz.-Nagy (see [12]) to spherical contractions: in general, spherical contractions have no spherical dilations, and there is not even a von Neumann-type inequality over the unit ball in Cn for spherical contractions ([3]). Athavale has shown in [1] that under certain additional positivity conditions a spherical contraction T has a spherical dilation, and M¨ uller and Vasilescu have developed a model for T under these conditions

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which reproduces this result ([6], [14]). This model consists of a spherical unitary part and a weighted backward multishift part which for suitable order coincides with the adjoint of the tuple of multiplication operators with the coordinates on a Hardy space over the unit ball in Cn . For n = 1, this is just the well-known coisometric extension for contractions. In the current paper, we will develop a model for a commuting tuple T under certain polynomial positivity conditions. We call T a P -contraction, where P P = aγ xγ is a polynomial with non-negative coefficients of a certain type, γ∈Nn P 0 ∗γ γ P if aγ T T 6 1H , and a P -unitary if aγ T ∗ γ T γ = 1H , T1 , . . . , Tn norγ∈Nn 0

γ∈Nn 0

mal. We will show that P -contractions satisfying additional positivity conditions of suitable order have a model consisting of a P -unitary part and a weighted backward multishift part, which may be identified topologically with the adjoint of the multiplication tuple on a Bergman space. In particular, up to topological equivalence, T has a P -unitary dilation and therefore a rich functional calculus. The crucial tools in identifying the weighted backward multishift with the adjoint Bergman space multiplication tuple are a theorem of A. Cumenge from complex analysis which allows to extend Bergman space functions on a complex submanifold M to Hardy space functions on a strictly pseudoconvex set containing M and the simple idea of regarding a P -contraction as a spherical contraction in a higher dimension. 2. PRELIMINARIES AND NOTATION

A commuting tuple T = (T1 , . . . , Tn ) of bounded linear operators on the separable Hilbert space H will be called a commuting multioperator or just a multioperator. For A ∈ L(H), let CA be the bounded linear map (2.1)

L(H) → L(H),

X 7→ A∗ XA,

and for a commuting tuple T = (T1 , . . . , Tn ) ∈ L(H)n let CT = (CT1 , . . . , CTn ). P If P = aγ xγ ∈ C[X1 , . . . , Xn ] is a polynomial, then P (CT ) is the bounded γ∈Nn 0 P linear map L(H) → L(H), X 7→ aγ T ∗ γ XT γ . This map is well-defined, since γ∈Nn 0

T1 , . . . , Tn commute. If T = (T1 , . . . , Tn ) is a commuting multioperator on H, S = (S1 , . . . , Sn ) a 0 0 commuting multioperator on some Hilbert space H and A : H → H is a linear map, then we will write AT = SA for the identity ATi = Si A, i = 1, . . . , n. In this situation, we call T and S topologically equivalent or similar if A is a

Standard models under polynomial positivity conditions

367

topological isomorphism. We will call a commuting multioperator normal in case all components are normal. For z = (z1 , . . . , zn ), w = (w1 , . . . , wn ) ∈ Cn , we will denote the tuple 2

2

(z 1 w1 , . . . , z n wn ) by zw and the tuple (|z1 | , . . . , |zn | ) by |z|2 . Let us introduce the class of polynomials from which our positivity conditions are obtained. A polynomial P ∈ C[X1 , . . . , Xn ] is said to be positive regular, if (i) the constant term is 0; (ii) P has non-negative coefficients; (iii) the coefficients of the linear terms X1 , . . . , Xn are all different from 0. There is a complete Reinhardt domain in Cn associated to each positive regular polynomial P , namely P = {z ∈ Cn | P (|z|2 ) < 1}

(2.2)

which we call the P -ball. For P =

n P

xi , the P -ball is just the unit ball Bn in Cn .

i=1

For a positive regular polynomial P, X ∈ L(H) positive and m ∈ N, we will call a commuting multioperator T (P, m)-positive for X, if (1)

∆P (X) := (1 − P )(CT )(X) > 0

(2.3) and

(m)

∆P (X) := (1 − P )m (CT )(X) > 0.

(2.4) In this case, (2.5)

(k)

∆P (X) := (1 − P )k (CT )(X) > 0

for 1 6 k 6 m,

as one obtains completely analogously to Lemma 2 in [6]. The tuple T is said to be (P, m)-positive, if it is (P, m)-positive for 1H . Furthermore, we call T a (1)

(1)

P -isometry, if ∆P := ∆P (1H ) = 0, and a P -unitary, if in addition T is normal. n P For P = xi , the (P, 1)-positive operators are just the spherical contraci=1

tions.

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3. STANDARD MODELS

We will now develop in analogy to [6] a standard model for (P, m)-positive commuting tuples, consisting of a part which is the adjoint of a multiplication tuple — or, equivalently, a weighted backward multishift — and a P -unitary part. For |P (x)| < 1, we have ∞ X m 1 j = P (x) . (1 − P (x))m j=0

(3.1)

Therefore the function x 7→ 1/(1 − P (x))m has a power series representation which converges compactly on {x |P (x)| < 1} and coincides with the Taylor series expansion at 0. For positive regular P , all Taylor coefficients are positive. Definition 3.1. Let P be a positive regular polynomial in n variables and let m ∈ N. For each α ∈ Nn0 , let ρm P (α) be the Taylor coefficient at index α of the m function x 7→ 1/(1 − P (x)) at 0. n We will denote the coefficients ρm P (α), α ∈ N0 , as (P, m)-weights. P 2 m cα z α such Now let H (ρP ) be the linear space of all formal power series n α∈N0 P 2 m |cα |2 1/ρm that P (α) < ∞. The space H (ρP ) is obviously a Hilbert space with α∈Nn 0

the inner product (3.2)

D X α∈Nn 0

cα z α ,

X

0

bα0 z α

E

=

α0 ∈Nn 0

X

cα bα

α∈Nn 0

1

. ρm P (α)

It can be regarded as a space of holomorphic functions on the P -ball P, and there is an obvious reproducing kernel: Lemma 3.2. The elements of H 2 (ρm P ) define holomorphic functions on the P -ball P. Furthermore, let (3.3)

k : P × P → C,

k(z, w) =

1 . (1 − P (zw))m

For each z ∈ P, the function kz = k(z, ·) is a holomorphic function on P and by identification with its Taylor series expansion at 0 an element of H 2 (ρm P ) such that hf, kz i = f (z), We have kkz k = 1/(1 − P (|z|2 ))m

1/2

f ∈ H 2 (ρm P ).

for z ∈ P.

369


P

Proof. For f =

α∈Nn 0

X

|cα z α | 6

α∈Nn 0

(3.4)

cα wα ∈ H 2 (ρm P ) and z ∈ P, we have

X

|cα |2

α∈Nn 0

1/2 1 1/2 X m α 2 ρ (α)|z | P ρm n P (α) α∈N0

1 = kf k. (1 − P (|z|2 ))m/2 Thus f converges uniformly on compact subsets of P and defines a holomorphic function on P (see [9], Corollaries 1.16 and 1.17), which we again call f . Furthermore, one obtains for z ∈ P

2

X X

α α |z α |2 ρm kkz k2 = ρm (α)z w

= P (α) P α∈Nn 0

α∈Nn 0

(3.5) = and hf, kz i =

1 0} and mult(P ) = |IP | be the number of nontrivial coefficients in P . We form the vector of the coefficients of P , A = (aγ )γ∈IP ∈ CIP . Furthermore, let for K = (kγ )γ∈IP , L = (lγ )γ∈IP ∈ CIP AK :=

(3.6)

Y

akγγ ,

|K| :=

γ∈IP

(3.7)

kγ ,

γ∈IP

|K| |K|! := Q , K kγ ! γ∈IP

X

Y lγ L := K kγ γ∈I P

and

(3.8)

[K] := ([K]1 , . . . , [K]n ),

where [K]i :=

X

γi kγ for i ∈ {1, . . . , n}.

γ∈IP

Write K 6 L if kγ 6 lγ for all γ ∈ IP . We need some combinatorial results:

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Sandra Pott

Lemma 3.3. For L ∈ NI0P and m ∈ N, X |L − K||K||K| + m − 1 |L| |L| + m . (3.9) = L−K K m−1 L m I K∈N0P K 6L

Proof. We obtain the identity X L |L| (3.10) = K r

for r = 0, . . . , |L|

K 6L |K|=r

by induction over the number of nontrivial coefficients |IP | of P and the well-known fact r X |L| − l l |L| (3.11) = for 0 6 l 6 |L|. q r−q r q=0 Now, we have X |L − K||K||K| + m − 1 L−K K m−1 I

K∈N0P K 6L

=

|L| X r=0

(3.12) =

"

# X |L| − r r r + m − 1 K m−1 L−K

K 6L |K|=r

|L| " # |L| X (|L| − r)! r! r + m − 1 X L L r=0 |L|! m−1 K K 6L |K|=r

=

|L| |L| X r + m − 1 . L r=0 m−1

It remains to show that

|L| P r=0

r+m−1 m−1

=

|L|+m m

for m ∈ N, which is an easy

induction. Furthermore, Equation (3.10) yields the identity X r |L| − |K| r!(|L| − r)! |L| X L |L| (3.13) = = K L−K |L|! L K L K 6L |K|=r

K 6L |K|=r

for 0 6 r 6 |L|. Now we can characterize the (P, m)-weights more explicitly.

371


Lemma 3.4. Let P be a positive regular polynomial and m ∈ N. Then (3.14)

ρm P (α)

X

=

K

A

I K∈N0P

|K| + m − 1 |K| |K| K

for α ∈ Nn0 .

[K]=α

Proof. For m = 1 and |P (x)| < 1, we have ∞

∞

X X 1 = P (x)j = 1 − P (x) j=0 j=0 (3.15)

∞ X

=

" X

j=0

K

# X |K| Y kγ γ kγ a (x ) K γ∈I γ I

"

K∈N0P |K|=j

A

I K∈N0P

|K| K

P

# [K]

x

X

=

α

x

α∈Nn 0

|K|=j

X I K∈N0P

K

A

|K| K

.

[K]=α

So, by uniqueness of the coefficients, (3.14) holds for m = 1. Now let (3.14) be valid for an arbitrary m ∈ N. Then we obtain again by uniqueness and by Lemma 3.3 the identity for m + 1 : X X 1 m α 1 α = ρP (α)x ρP (α)x (1 − P (x))m+1 α∈Nn α∈Nn 0 0 X X |K| |K| + m − 1 |J| = AK x[K] AJ x[J] K m−1 J I I K∈N0P

(3.16)

J∈N0P

" =

X

L [L]

A x

I L∈N0P

# X |L − K| |K| + m − 1 |K| L−K m−1 K I

K∈N0P K 6L

" =

X α∈Nn 0

α

x

X

L

A

I L∈N0P

|L| L

|L| + m m

# .

[L]=α

n n Let from now on ρm P (α) = 0 for α ∈ Z \ N0 . Then we obtain the following recursion formulae for the (P, m)-weights: P Remark 3.5. Let P = aγ xγ be a positive regular polynomial and let γ∈Nn 0 P Q = 1 − (1 − P )m = bγ xγ . Then γ∈Nn 0

(3.17)

ρm P (α) =

X γ∈Nn 0

b γ ρm P (α − γ),

α ∈ Nn0

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Sandra Pott

and for m > 1, m−1 ρm (α) + P (α) = ρP

(3.18)

X

aγ ρm P (α − γ).

γ∈Nn 0

P Proof. For α ∈ Nn0 , bγ ρm P (α − γ) is the coefficient at index α of the γ∈Nn 0 P P α product power series ρm bγ xγ . We obtain Equation (3.17) P (α)x α∈Nn 0

γ∈Nn 0

by comparison of coefficients, since for |P (x)| < 1 we have X (3.19)

xα

α∈Nn 0

X

−m b γ ρm (1 − (1 − P (x))m ) P (α − γ) =(1 − P (x))

γ∈Nn 0

=

X

α ρm P (α)x − 1.

α∈Nn 0

P Similarly, aγ ρm P (α − γ) is the α-coefficient of the product power series γ∈Nn 0 P P α γ ρm (α)x a x , and we obtain for |P (x)| < 1, m > 1 γ P α∈Nn 0

γ∈Nn 0

X

X xα ρm aγ ρm P (α) − P (α − γ) − 1

α∈Nn 0

γ∈Nn 0

= (1 − P (x))−m − (1 − P (x))−m P (x) − 1 X = (1 − P (x))−m+1 − 1 = ρm−1 (α)xα − 1 P

(3.20)

α∈Nn 0

implying (3.18). Now we can prove that the multiplication operators are well-defined bounded operators on H 2 (ρm P ). Lemma 3.6. Mz1 , . . . , Mzn ∈ L(H 2 (ρm P )). Proof. Let ei be the ith unit vector in Cn , i = 1, . . . , n. It is sufficient to m n show that for some constant c > 0, ρm P (α + ei ) > cρP (α) for all α ∈ N0 . But by

Remark 3.5, (3.21)

ρm P (α + ei ) >

X

m aγ ρm P (α + ei − γ) > aei ρP (α)

γ∈Nn 0

for α ∈ Nn0 , which proves the lemma.

373


The multiplication operators are obviously commuting. For the separable Hilbert space H, we can consider the Hilbert space tensor 2 m product H ⊗ H 2 (ρm P ) =: HH (ρP ). This space can obviously be identified with the P space of formal power series with coefficients in H, hα z α with hα ∈ H for α∈Nn 0 P 2 m α ∈ Nn0 , such that khα k2 (1/ρm P (α)) < ∞. The inner product on HH (ρP ) is α∈Nn 0

then given by D X

hα z α ,

α∈Nn 0

X

0

h0α0 z α

E

=

X

hhα , h0α i

α∈Nn 0

α0 ∈Nn 0

1

. ρm P (α)

2 We can view HH (ρm P ) as a space of H-valued holomorphic functions on P. From

now on, we will denote the multiplication operators with the coordinates on 2 2 m HH (ρm P ) as well as the ones on H (ρP ) by Mz1 , . . . , Mzn . By Lemma 3.6, these 2 operators are also well-defined and bounded on HH (ρm P ).

As in the case of spherical contractions, the spectrum of a (P, 1)-positive multioperator is contained in the closure of the P -ball: Lemma 3.7. Let P be a positive regular polynomial and T a (P, 1)-positive commuting multioperator. Then the Taylor spectrum σ(T ) of T is contained in the closure P of the P -ball. Proof. This lemma is a special case of a more general result ([11], Theorem 1.12). We give a more elementary proof for our situation. Let λ ∈ Cn \ P. We will show that λ is not contained in the joint spectrum of T relative to the closed commutative subalgebra A of L(H) generated by T1 , . . . , Tn , i.e. we will show that the ideal I generated by λ1 1H −T1 , . . . , λn 1H −Tn in A is equal to A. Since the Taylor spectrum σ(T ) of T is contained in the joint spectrum of T relative to any closed commutative subalgebra of L(H) containing T , this means that λ is not in σ(T ). Let Qλ (z) = (1/P (|λ|2 ))P (λz). Then Qλ (λ) = 1, and for h ∈ H, khk 6 1, 1 kP (λT )hk P (|λ|2 ) 1/2 X 1/2 1 X γ 2 γ 2 6 a |λ | a kT hk γ γ P (|λ|2 )

kQλ (T )hk = (3.22)

γ∈IP

γ∈IP

1 1 = hP (CT )(1H )h, hi1/2 6 0 and consequently converging to some positive operator PeX in the SOT-topology. Now define for X ∈ L(H), X > 0, and T (P, m)-positive for X the map X 1/2 α 2 m V1X : H → HH (ρm h 7→ ρm T hz α . P ), P (α) ((1 − P (CT )) (X)) α∈Nn 0

As one proves by induction completely analogously to [6], Lemmas 4 and 5 (see also [11], 2.1 and 2.8), we have k X j+m−1 (3.24)

j=0

m−1

P (CT )j (1 − P (CT ))m

= 1−

m−1 X j=0

k+j P (CT )k+1 (1 − P (CT ))j , j

k∈N

375


and (3.25)

lim

k→∞

k+j hP (CT )k+1 (1 − P (CT ))j (X)h, hi = 0, j

h ∈ H,

for j = 1, . . . , m − 1. We obtain (3.26)

kV1X hk2 = khk2 − lim hP (CT )k (X)h, hi = khk2 − hPeX h, hi,

h∈H

k→∞

by kV1X hk2 =

X

m α α ρm P (α) h(1 − P (CT )) (X)T h, T hi

α∈Nn 0

" X

=

α∈Nn 0

= (3.27)

∞ X

K∈N0P [K]=α

"

j=0

=

# X |K| + m − 1|K| K α m A hCT (1 − P (CT )) (X)h, hi m−1 K I

# D E X j + m − 1 j [K] K m A CT (1 − P (CT )) (X)h, h m−1 K I

K∈N0P |K|=j

∞ X j+m−1 m−1

j=0 2

= khk − lim

k→∞

hP (CT )j (1 − P (CT ))m (X)h, hi

m−1 X j=0

k+j hP (CT )k+1 (1 − P (CT ))j (X)h, hi j

2

= khk − lim hP (CT )k (X)h, hi, k→∞

according to (3.24) and (3.25), with the limits existing because of Claim 1. For T (P, m)-positive and V1 = V11H , one gets V1 T i h =

X

m 1/2 α+ei ρm T h zα P (α)((1 − P (CT ) (1H ))

α∈Nn 0

=

X α∈Nn 0

(3.28) =

Mz∗i

ρm P (α) ρm (α m ρP (α + ei ) P

X

ρm P (α

+ ei )((1 − P (CT ))m (1H ))1/2 T α+ei h z α m

1/2

+ ei )((1 − P (CT )) (1H ))

T

α+ei

hz

α+ei

α∈Nn 0

= Mz∗i V1 h. So we have constructed the first part of our model. In a second step we construct the P -unitary part, using the fact that Pe = Pe1H is invariant under P (CT ). In the following, we write s - lim for the limits in the strong operator topology on L(H).

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Sandra Pott

Lemma 3.9. Let T be a (P, 1)-positive commuting multioperator on H and e e P = P1H = s - lim P (CT )k (1H ). Then there exist a Hilbert space N , a P -unitary k→∞

multioperator N ∈ L(N )n and a contractive linear mapping V2 : H → N such that kV2 hk2 = hPeh, hi for h ∈ H and V2 T = N V2 . Proof. Let K = Pe1/2 H and V2 : H → K, h 7→ Pe1/2 h. For i = 1, . . . , n, the linear map Wi : Pe1/2 H → K, (3.29)

Wi V2 h = V2 Ti h

for h ∈ H,

is well-defined and bounded, since −1 2 e (3.30) kWi V2 hk2 = hTi∗ Pe Ti h, hi 6 a−1 ei hP (CT )(P )h, hi = aei kV2 hk ,

h ∈ H.

So we can extend Wi to a bounded linear map K → K, which we also call Wi . By (3.29) and continuity, we have W V2 = V2 T for W = (W1 , . . . , Wn ) and consequently (3.31)

V2∗ (P (CW )(1K ))V2 = P (CT )(V2∗ V2 ) = P (CT )(Pe) = V2∗ V2

because of the SOT-continuity of P (CT ). Now P (CW )(1K ) = 1K , since V2 H is dense in K. Thus W is a P -isometry. To replace W by a P -unitary tuple, we need the following lemma: Lemma 3.10. Every P -isometry is subnormal, and its minimal normal extension is a P -unitary. 1/2

Proof. Let W ∈ L(W)n be a P -isometry. Then the tuple (aγ W γ )γ∈IP is a spherical isometry and consequently by [1], Proposition 2, a subnormal tuple. Since ae1 , . . . , aen are all not 0, in particular the tuple W = (W1 , . . . , Wn ) is subnormal. Let N = (N1 , . . . , Nn ) be its minimal normal extension on the Hilbert 1/2 space N ⊇ K. Then (aγ N γ )γ∈IP is the minimal normal extension of the tuple 1/2 (aγ W γ )γ∈IP and by [1] also a spherical isometry, which implies that N is a P -unitary. Now let for a (P, m)-positive multioperator T on H (3.32)

2 V = V 1 ⊕ V 2 : H → HH (ρm P ) ⊕ N.

377


The mapping V is an isometry, and V T = (Mz∗ ⊕ N )V . Note that only the first part of the model depends on m. For the proof of the reverse direction, we have only to show that Mz∗ ∈ n ∗ L(H 2 (ρm P )) is (P, m)-positive for arbitrary m. Then the (P, m)-positivity of Mz 2 m on HH (ρP ) follows, and we obtain the (P, m)-positivity of T by the fact that any P -unitary is (P, m)-positive for every m and that (P, m)-positivity is preserved under the direct sum Mz∗ ⊕ N , the restriction to the invariant subspace V H and the unitary transformation H → V H. Lemma 3.11. For every m ∈ N, the commuting multioperator Mz∗ ∈ n m L(H 2 (ρm P )) is (P, m)-positive. Moreover, (1 − P (CMz∗ )) (1) is the orthogonal projection onto the subspace of constants in H 2 (ρm P ). Proof. For α, β ∈ Nn0 , we have ( Mzβ Mz∗ β z α

(3.33)

=

ρm P (α−β) α ρm (α) z P

if β 6 α,

0

otherwise.

So obviously (1 − P (CMz∗ ))m (1)z α = z α for α = 0. Let as before ρm P (α) = 0 n n ∗β α β for α ∈ Z \ N0 . Since the spaces C · z are invariant under Mz Mz , thus also invariant under (1 − P (CMz∗ ))(1) and (1 − P (CMz∗ ))m (1), it remains to show that h(1 − P (CMz∗ ))(1)z α , z α i > 0,

(3.34)

m

α

α

h(1 − P (CMz∗ )) (1)z , z i = 0,

(3.35)

α>0 α > 0, α 6= 0.

By Equation (3.33), we have h(1 − P (CMz∗ ))(1)z α , z α i =

(3.36)

X 1 m m ρ (α) − a ρ (α − γ) γ P P 2 ρm P (α) γ∈IP

and (3.37)

h(1 − P (C

Mz∗

X 1 m m bγ ρP (α − γ) , )) (1)z , z i = m 2 ρP (α) − ρP (α) n m

α

α

γ∈N0

where

P

bγ xγ is the polynomial 1 − (1 − P )m . The rest of the proof now results

γ∈Nn 0

from Remark 3.5. This finishes the proof of Theorem 3.8.

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Sandra Pott

Via the isometric isomorphism 2 2 n HH (ρm P ) → l (N0 , H),

(3.38)

X α∈Nn 0

hα z α 7→

1 h , 1/2 α α∈Nn ρm 0 P (α)

the multioperator Mz∗ may be looked upon as a weighted multi-backward shift. So 2 V 1 H ⊆ HH (ρm P ) may be regarded as the shift part of our model, and V2 H ⊆ N is the P -unitary part. In case m = n = 1 and P = x, the (P, m)-positive operators are just the contractions, and our model is the well-known coisometric extension for contractions. n P If P is the polynomial xi , the P -ball P = {z ∈ Cn | P (|z|2 ) < 1} is just i=1

the unit ball Bn of Cn , and the P -unitaries are just the spherical unitaries. For this case, Theorem 3.8 was proved by V. M¨ uller and F.-H. Vasilescu in [6]. The (m) positivity conditions ∆P > 0, 1 6 m 6 n, were examined earlier by A. Athavale, who showed in [1], Remark 1 to Proposition 4, that the tuple T then has a spherical dilation. The standard model of M¨ uller and Vasilescu reproduces this result: as one easily verifies, for the above P the space H 2 (ρm P ) is just the Hardy space Z n 2 2 H (B ) = f : B → C holomorphic kf k := sup |f (rz)| dσ < ∞ , 2

n

0 0} and |IP | = mult(P ) = m, identify C with C and denote the elements of Cm by w = (wγ )γ∈IP . Let τ : Cm → Cn , w = (wγ )γ∈IP 7→ −1/2 −1/2 (we1 , . . . , wen ), and κ : Cm → Cn , w = (wγ )γ∈IP 7→ (ae1 we1 , . . . , aen wen ). Now define the holomorphic map ( 1/2 aγ wγ if γ ∈ e1 , . . . , en , m m (4.1) ϕ : C → C , ϕ(w)γ = 1/2 wγ + aγ τ (w)γ otherwise. The map ϕ is biholomorphic, since ( (4.2)

ϕ

−1

:C

m

m

→C ,

ϕ

−1

(w)γ =

−1/2

aγ

wγ

if γ ∈ e1 , . . . , en ,

wγ −

1/2 aγ κ(w)γ

otherwise;

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Sandra Pott

is obviously a holomorphic inverse map. Let D = ϕ−1 (Bm ). Then D is strictly pseudoconvex, since Bm is strictly pseudoconvex (see e.g. [9], II.2.7), and we have D ∩ (Cn × {0} × · · · × {0}) X aγ |τ (w)γ |2 < 1 / {e1 , . . . , en }, = w ∈ Cm wγ = 0 for γ ∈

(4.3)

γ∈IP

=P × {0} × · · · × {0}. Moreover, M = ϕ(P) is a complex submanifold of Bm such that M = {w ∈ Bm | 1/2

wγ = aγ κ(w)γ }. Let Q be the polynomial in m variables that corresponds to the unit ball, P Q ∈ C[(Xγ )γ∈IP ], Q = xγ . γ∈IP

We will now construct the identifying map B 2 (P, µ) → H 2 (ρm P ) in several steps. IP n Step 1. The restriction. As in (3.8), let [ · ] : Nm 0 = N0 → N0 , [β]i =

P

γi βγ .

γ∈IP

Lemma 4.2. With A = (aγ )γ∈IP and the notation in (3.6), the map (4.4)

X

π : H 2 (Bm ) → H 2 (ρm P ),

cβ wβ 7→

β∈Nm 0

X

cβ Aβ/2 z [β]

β∈Nm 0

is well-defined, surjective, linear and has norm 1. Proof. First notice that the (P, m)-weights may be expressed in terms of (Q, m)-weights: For α ∈ Nn0 , we have (4.5)

ρm P (α)

=

X β∈Nm 0 [β]=α

β

A

X |β| + m − 1 |β| = Aβ ρm Q (β). m−1 β m β∈N0 [β]=α

As one shows easily by induction over r, for any a1 , . . . , ar , b1 , . . . , br ∈ R with a1 , . . . , ar > 0 and b1 , . . . , br > 0 one has P r (4.6)

ai

i=1 r P i=1

2 6

bi

r X a2 i

i=1

bi

.

381


Consequently we obtain for arbitrary f =

P β∈Nm 0

2 P β/2 A c β m (4.7)

P

β∈N0 [β]=α ρm P (α)

6

2

X

β∈Nm 0 [β]=α

P β∈Nm 0 [β]=α

cβ wβ ∈ H 2 (Bm ), α ∈ Nn0

2 Aβ/2 |cβ | Aβ ρm Q (β)

6

X β∈Nm 0 [β]=α

|cβ |2 ρm Q (β)

and (4.8)

kπ(f )k =

α∈Nn 0

2 1 X β/2 A cβ 6 kf k2 . ρm m P (α) β∈N0 [β]=α

2 m To show the surjectivity of π, consider the map ι : H 2 (ρm P ) → H (B ), g = P P P α β/2 m m β cα z 7→ cα A (ρQ (β)/ρP (α))w . Then ι is well-defined and iso-

α∈Nn 0

α∈Nn 0

β∈Nm 0 [β]=α 2 m

metric, since ι(g) ∈ H (B ) with kι(g)k2 =

P α∈Nn 0

|cα |2

P β∈Nm 0 [β]=α

2 m Aβ (ρm Q (β)/ρP (α) ) =

kgk2 by Equation (4.5), and π ◦ ι = 1. Thus the map π can be regarded as the orthogonal projection from H 2 (Bm ) 2 m onto the closed subspace H 2 (ρm P ). This close relationship between H (ρP ) and H 2 (Bm ) and the definitions of ϕ and π become clearer by considering the following idea: Let T = (T1 , . . . , Tn ) be a (P, m)-positive multioperator on H and let V1 : 2 (ρm H → HH P ) be the map constructed in Theorem 3.8. Let W be the commuting 1/2 m-tuple (Wγ )γ∈IP , Wγ = aγ T γ . Then (1 − P )(CT ) = (1 − Q)(CW )

(4.9)

and thus W is (Q, m)-positive. Again by Theorem 3.8, now applied to the m-tuple 2 (Bm ) as first part of the model for the tuple W , we obtain the map Ve1 : H → HH W . Therefore X m 1/2 (1H ⊗π) ◦ Ve1 (h) = (1H ⊗π) ρm W β hwβ Q (β)((1−Q) (CW )(1H )) β∈Nm 0

(4.10)

=

X X

β m 1/2 [β] ρm T hz α Q (β)A ((1 − P ) (CT )(1H ))

m α∈Nn 0 β∈N0

[β]=α

=

X α∈Nn 0

m 1/2 α ρm T hz α = V1 (h) P (α)((1 − P ) (CT )(1H ))

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for h ∈ H, and we have (1H ⊗ π) ◦ Ve1 = V1 .

(4.11)

In particular, the map 1H ◦ π is isometric on Ve1 H, since (4.12)

kV1 hk2 = lim hP (CT )k (1H )h, hi = lim hQ(CW )k (1H )h, hi = kVe1 hk2 . k→∞

k→∞

1/2 The submanifold M = w ∈ Bm wγ = aγ κ(w)γ corresponds to the identities 1/2 Wγ = aγ T γ . The map π may be regarded as the restriction of functions in H 2 (Bm ) to the submanifold M, up to the biholomorphic map ϕ. For z ∈ P and P f= cβ wβ ∈ H 2 (Bm ), we have β∈Nm 0

X

f ◦ ϕ(z) =

cβ (ϕ(z))β =

β∈Nm 0

(4.13)

X

=

X

Y

cβ

β∈Nm 0 β/2 [β]

cβ A

z

aβγ γ /2 (z γ )βγ

γ∈IP

= π(f )(z).

β∈Nm 0

Altogether, we have the following commutative diagram. Ve1 H   e1 % V yo

(4.14) H

V

1 −→

V1 H

,→

,→

2 (Bm ) HH x  1 ⊗ ιy1H ⊗ π = · ◦ ϕ|P 2 (ρm HH P ).

Step 2. The transformation. Recall that the Hardy space H p (Ω), 1 < p < ∞, over a bounded strictly pseudoconvex set Ω ⊆ Cn with C 2 -boundary can be obtained in the following way (see e.g. [5], Section 8.3): Let % : U → R be a strictly plurisubharmonic defining C 2 -function for Ω, defined on some region U ⊃ Ω. That means, Ω = {z ∈ U | %(z) < 0}.

(4.15)

Now for ε > 0 let Ωε = {z ∈ U | %(z) < ε}. For sufficiently small ε0 , ∂Ωε is a real C 2 -manifold for each ε with 0 < ε < ε0 . Let σε be the surface measure on ∂Ωε and define (

(4.16) H (Ω) = f : Ω → C holomorphic kf kp = p

Z sup ε0 >ε>0 ∂Ωε

!1/p p

|f (z)| dσε

) |f |p }

1/p

defines an equivalent norm to k · kp on H p (Ω) (see e.g. [15], Section 2.2). Since composition with the biholomorphic map ϕ maps the class of realvalued harmonic functions on Bm bijectively onto the class of real-valued harmonic functions on D, for any fixed z0 ∈ D and any f ∈ H 2 (Bm ) we have kf ◦ ϕk22,z0 = inf{g(z0 ) | g : D → R harmonic, g > |f ◦ ϕ|2 } (4.20)

= inf{g(ϕ(z0 )) | g : Bm → R harmonic, g > |f |2 } = kf k22,ϕ(z0 ) ,

and Uϕ in (4.17) is thus a topological isomorphism with inverse Uϕ−1 . Step 3. The extension. Now we come to the main step of our construction of the identification B 2 (P, µ) → H 2 (ρm P ), using a theorem of A. Cumenge. We will show that for a measure µ e equivalent to µ, there is a bounded 2 linear extension operator E : B (P, µ e) → H 2 (D) and that the restriction R : H 2 (D) → B 2 (P, µ e) is well-defined, bounded and surjective. To apply the theorem of Cumenge, we first have to show that P may be extended to a complex manifold e of Cm intersecting transverse to ∂D, i.e. that there is a complex submanifold P e ∂D transversally such that P = D ∩ P.

384

Sandra Pott

e = Cn × {0} × · · · × {0}. Then P = D ∩ P e by (4.3). The function Let P P m 2 r : C → R, r(z) = |zγ | − 1, is a strictly plurisubharmonic defining C ∞ γ∈IP

function for Bm . Thus % = ϕ◦r is a strictly plurisubharmonic defining C ∞ -function for D. e intersects ∂D transversally, we have to show that To prove that P ! (4.21)

^

d%(z) ∧

6= 0

dzγ

e ∩ ∂D for all z ∈ P

γ∈IP \{e1 ,...,en }

e ∩ ∂D, there is an (see e.g. [9], p. 118). So it suffices to prove that for every z ∈ P e identify z with ze = τ (z) ∈ Cn to i ∈ {1, . . . , n} such that ∂%/∂zei (z) 6= 0. On P, P γ 2 e obtain %(z) = aγ |z | . Now let z ∈ P ∩ ∂D. Since 0 ∈ / ∂D, there is an i with γ∈IP

τ (z)i 6= 0, and we obtain ∂% ∂% (z) = (e z ) = ae1 τ (z)i + ∂zei ∂e zi (4.22)

X

γi aγ τ (z)γ τ (z)γ−ei

γ∈IP \{e1 ,...,en } γi 6=0

! X

= τ (z)i aei +

γ−ei 2

γi aγ |τ (z)

|

6= 0,

γ∈IP \{e1 ,...,en } γi 6=0

since the second factor is strictly positive. Now P is a complex submanifold of codimension m − n of the smoothly bounded strictly pseudoconvex set D. Thus we are in the situation of Theorem 0.1 in [2]: let µ e be the measure dist(z, ∂D)dλ on P. Then f |P ∈ B 2 (P, µ e) for every 2 f ∈ H (∂D), and there exists a bounded linear extension operator E : B 2 (P, µ e) → H 2 (D), Eg|P = g for g ∈ B 2 (P, µ e). Moreover, the restriction operator R : H 2 (D) → B 2 (P, µ e) is bounded since µ e is a Carleson measure on D by H¨ormander’s formulation of Carleson’s Theorem and by Lemme II.1.1 in [2] (see [2], Section II.1, and [4], Theorem 4.3). It is surjective since R ◦ E = 1B 2 (P,˜µ) . The map π ◦ Uϕ−1 ◦ E : B 2 (P, µ e) → H 2 (ρm P) now maps each function g ∈ B 2 (P, µ e) onto itself. It is bounded by construction and has the bounded inverse R ◦ Uϕ ◦ ι. Altogether, we have the following commutative diagram: H 2 (D) x  E yR e) B 2 (P, µ

Uϕ−1

−→

H 2 (B)m x  y

∼

H 2 (ρm P)

−→ id

385


It remains to compare µ and µ e. Step 4. The equivalence of the measures. It suffices to show that there are constants c1 , c2 > 0 such that (4.23)

2

2

c1 dist(z, ∂D) 6 1 − P (|z1 | , . . . , |zn | ) 6 c2 dist(z, ∂D),

z ∈ ∂P.

Then B 2 (P, µ) and B 2 (P, µ e) coincide as sets and carry equivalent norms. The second inequality just follows by the Lipschitz continuity of the map 2 2 z 7→ P (|z1 | , . . . , |zn | ) on the compact set P. For the first inequality, choose for z ∈ P some w ∈ ∂P such that z = λw for a suitable λ ∈ [0, 1). Then X 2 2 aγ (|wγ |2 − |z γ |2 ) 1 − P (|z1 | , . . . , |zn | ) = γ∈IP

(4.24)

> (1 − λ2 )

n X

aei |wi |2 > c(1 − λ)kwk2

i=1

> c1 (1 − λ)kwk = c1 kw − zk > c1 dist(z, ∂P) for suitable constants c, c1 > 0, since ∂P is bounded away from 0. Thus we obtain (4.23), which finishes the proof of the theorem.

5. DILATIONS

The identifying map B 2 (P, µ) → H 2 (ρm P ) obviously intertwines the multiplication operators with the coordinate functions on B 2 (P, µ) and H 2 (ρm P ). So its adjoint intertwines the adjoints of the multiplication operators, and we obtain the following easy consequence of Theorem 3.8 and Theorem 4.1. Let as before P be a positive regular polynomial with m = mult(P ) > n, µ the normalization of the measure 2 2 (1 − P (|z1 | , . . . , |zn | ))m−n−1 dλ on P and let M = (M1 , . . . , Mn ) be the tuple of 2 multiplication operators with the coordinate functions on BH (P, µ). Corollaty 5.1. The following are equivalent: (i) T is topologically equivalent to a (P, m)-positive multioperator; 2 (ii) T is topologically equivalent to the restriction of M ∗ ⊕N ∈ L(BH (P, µ)⊕ N )n to an invariant subspace, where N is a P -unitary operator on some separable Hilbert space N . Moreover, the functional model for a (P, m)-positive multioperator T implies — up to topological equivalence — the existence of a P -unitary dilation for T . Unlike the situation of the unit ball, we cannot obtain a P -unitary dilation directly. We have to check the complete boundedness of the map q 7→ q(T ) on the algebra of polynomials, equipped with the supremum norm on P.

386

Sandra Pott

Theorem 5.2. Let T be a (P, m)-positive commuting multioperator. Then T is topologically equivalent to a multioperator S which has a P -unitary dilation. Proof. By Corollary 5.1, T is topologically equivalent to the restriction of M ⊕ N to an invariant subspace. Thus it is sufficient to show that M ∗ has a P -unitary dilation. The algebra C[X1 , . . . , Xn ] carries an operator algebra structure as a subalgebra of the commutative C ∗ -algebra C(∂P) of continuous functions on ∂P. We denote this operator algebra by Pol(P). ∗

Remark 5.3. The algebra homomorphism (5.1)

2 Φ : Pol(P) → L(BH (P, µ)),

q 7→ q(M ∗ )

is completely contractive. 2 Proof. Let Mn (L(BH (P, µ))) be the C ∗ -algebra of n × n-matrices over 2 (P, µ)) and let Mn (Pol(P)) be the algebra of n × n-matrices over Pol(P), L(BH carrying the norm k(qi,j )kn = sup{k(qi,j (z))k z ∈ P}, where k(qi,j (z))k denotes the usual operator norm of the complex n × n-matrix (qi,j (z)). We have to show that for each n, the map

(5.2)

2 Φ(n) : Mn (Pol(P)) → Mn (L(BH (P, µ))),

(qi,j ) 7→ (qi,j (M ∗ ))

is a contraction. ∨ For q ∈ C[X1 , . . . , Xn ], let q be the polynomial obtained by complex conjugation of the coefficients of q. Then for (qi,j ) ∈ Mn (Pol(P)), kΦ(n) ((qi,j ))k = ∨ n 2 2 (P, µ) = BH k(qi,j (M ∗ ))k = k(q j,i (M ))k, and for f = (f1 , . . . , fn ) ∈ BH n (P, µ) we have Z Z

2

∨ ∨ ∨ 2 2 q q 2 (P,µ) )f (z) dµ k( j,i (M ))f k = k(( j,i (M ))f )(z)k dµ = (q j,i (z)1BH (5.3)

P Z ∨ ∨ 2 2 6 k(q j,i (z))k kf (z)k dµ 6 k(q j,i )k2n kf k2 = k(qi,j )k2n kf k2 . P

P

Thus Φ(n) is a contraction, and the remark is proved. To finish the proof of the theorem, note that by a corollary to Arveson’s Extension Theorem (see [7], Corollary 6.7) the map Φ dilates to a homomorphism 2 Ψ : C(P) → L(K) with some Hilbert space K ⊇ BH (P, µ). Then the tuple K = (Ψ(z1 ), . . . , Ψ(zn )) is a normal multioperator dilating M ∗ , and the Taylor spectrum

387


of K is contained in ∂P. By the Spectral Theorem for normal multioperators (see [13], Theorem 7.26), we have Z (5.4)

P (CK )(1K ) =

P (|z|2 ) dE = 1K ,

∂P

where E is the spectral measure for the tuple K on K. In particular, Theorem 5.2 implies that each (P, m)-positive multioperator satisfies a von Neumann-type inequality with respect to the P -ball P. Let A(P) be the Banach algebra of complex-valued continuous functions on P which are holomorphic on P, together with the supremum norm on P. Corollary 5.4. Let T be a (P, m)-positive multioperator. Then T has a continuous A(P)-functional calculus. In particular, there is a constant c > 0 such that kq(T )k 6 c sup |q(z)| z ∈ P

(5.5)

for q ∈ C[X1 , . . . , Xn ].

Proof. As one easily sees by the Spectral Theorem for normal multioperators (see [13], Theorem 7.26) and by Lemma 3.7, a P -unitary multioperator U satisfies the von Neumann-inequality kq(U )k 6 sup |q(z)| z ∈ P

(5.6)

for q ∈ C[X1 , . . . , Xn ].

The corollary now follows from Theorem 3.8, since the polynomials are dense in A(P). In case the model for T provided by Theorem 5.2 consists only of the multiplication operator part, i.e. in case P (CT )s (1H ) converges strongly to 0 for s → ∞, 2 2 (ρm we can strengthen this result. Let A : HH P ) → BH (P, µ) be the isomorphism ∗ 2 m ∗ 2 intertwining Mz on HH (ρP ) and M on BH (P, µ) mentioned in the beginning of this paragraph. Then H ∞ (P) → L(H),

(5.7)

f 7→ V ∗ A−1 Mfˇ∗ AV, ∨

2 where V : H → HH (ρm P ) is the isometry constructed in Theorem 3.8, f is the holomorphic map z 7→ f (z) on P and Mfˇ is the bounded operator of multiplication ∨

2 with f on BH (P, µ), defines a continuous algebra homomorphism with norm less or equal to kAk kA−1 k, mapping the coordinate functions to the components of T . Thus (5.7) gives a continuous H ∞ (P)-functional calculus for T .

388

Sandra Pott

In a forthcoming paper ([9]), the developed standard model for (P, m)positive multioperators T will be applied to give necessary conditions for the existence of non-trivial joint invariant subspaces of T .

Acknowledgements. This paper constitutes part of the author’s Ph.D. Dissertation written at the University of Saarbr¨ ucken under the direction of Prof. Dr. Ernst Albrecht. I would like to thank Prof. Albrecht for many valuable discussions and suggestions.

REFERENCES

1. A. Athavale, On the intertwining of joint isometries, J. Operator Theory 23(1990), 339–350. 2. A. Cumenge, Extension dans des classes de Hardy de fonctions holomorphes et estimations de type “mesures de Carleson” pour l’equation ∂, Ann. Inst. Fourier (Grenoble) 33(1983), 59–97. 3. S.W. Drury, A generalization of von Neumann’s inequality to complex ball, Proc. Amer. Math. Soc. 68(1978), 300–304. ¨ rmander, Lp -Estimates for (pluri-)subharmonic functions, Math. Scand. 20 4. L. Ho (1967), 65–78. 5. S.G. Krantz, Function Theory of Several Complex Variables, Pure and Applied Mathematics Series, John Wiley and Sons Inc., New York 1982. ¨ ller, F.-H. Vasilescu, Standard models for some commuting multioperators, 6. V. Mu Proc. Amer. Math. Soc. 117(1993), 979–989. 7. V. Paulsen, Completely Bounded Maps and Dilations, Pitman Res. Notes Math. Ser., vol. 146, Longman, London 1986. 8. S. Pott, Invariant subspaces for tuples with normal dilations, in Banach Algebra, Walter DeGruyter, Berlin 1998, pp. 397–413. 9. M.R. Range, Holomorphic Functions and Integral Representations in Several Complex Variables, Grad. Texts in Math., vol. 108, Springer-Verlag, New York 1986. 10. W. Rudin, Function Theory in Unit Ball of Cn , Grundlehren Math. Wiss., vol. 241, Springer Verlag, New York 1980. 11. J. Schult, Modelle kommutierender Operatoren, Diplomarbeit, University of Saarbr¨ ucken 1994. 12. B. Sz.-Nagy, C. Foias¸, Harmonic Analysis on Hilbert Spaces, North Holland Publishing Company, Amsterdam 1970. 13. F.-H. Vasilescu, Analytic Functional Calculus and Spectral Decompositions, D. Reidel Publishing Company, Dordrecht 1982.


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14. F.-H. Vasilescu, An operator-valued Poisson kernel, J. Funct. Anal. 110(1992), 47–72. 15. R. Wolff, Spectral theory on Hardy spaces in several complex variables, Ph.D. Dissertation, Westf¨ alische Wilhelms-Universit¨ at M¨ unster 1996.

SANDRA POTT Fachbereich Mathematik Universit¨ at des Saarlandes D–66041 Saarbrucken GERMANY Current address: Department of Mathematics and Statistics University of Edinburgh Edinburgh EH9 3JZ U.K. E-mail: [email protected] Received June 24, 1997.