Functional models of Γn -contractions and characterization of Γn -isometries

Journal of Functional Analysis 266 (2014) 6224–6255

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Functional models of Γn -contractions and characterization of Γn -isometries Shibananda Biswas ∗,1 , Subrata Shyam Roy Indian Institute of Science Education and Research, Pincode 741252, Nadia, West Bengal, India

a r t i c l e

i n f o

Article history: Received 18 November 2013 Accepted 20 February 2014 Available online 26 March 2014 Communicated by G. Schechtman Keywords: Symmetrized polydisc Spectral set Beurling–Lax–Halmos theorem Symmetric group Irreducible representations

a b s t r a c t We show that a weighted Bergman space on the polydisc splits up into the orthogonal direct sum of subspaces, each corresponding to an irreducible representation of the symmetric group. The multiplication by the n-tuple of elementary symmetric functions is a Γn -contraction on each of these subspaces. A necessary condition for a commuting n-tuple to be a Γn -contraction is obtained in terms of pencil of operators. We provide a model theory for Γn -isometries. A Beurling–Lax–Halmos type representation for the invariant subspaces of Γn -isometries is given. © 2014 Elsevier Inc. All rights reserved.

1. Introduction We denote by D and D the open and closed unit discs in the complex plane C, respectively. Let si , i  0, be the elementary symmetric function in n variables of degree i, that is, si is the sum of all products of i distinct variables zi so that s0 = 1 and  si (z) = zk1 · · · zki , 1k1
* Corresponding author. E-mail addresses: [email protected] (S. Biswas), [email protected] (S. Shyam Roy). The work of S. Biswas was supported in part by Inspire Faculty Award [IFA-11MA-06] funded by DST, India at Indian Statistical Institute, Kolkata. 1

http://dx.doi.org/10.1016/j.jfa.2014.02.034 0022-1236/© 2014 Elsevier Inc. All rights reserved.

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where z = (z1 , . . . , zn ). For n  1, let s : Cn −→ Cn be the function of symmetrization given by the formula   s(z) = s1 (z), . . . , sn (z) . The image Γn := s(Dn ) under the map s of the unit n-polydisc is known as the symmetrized n-disc or simply symmetrized polydisc. The map s is a proper holomorphic map [16]. As in [2], any commuting n-tuple of operators having Γn as a spectral set will be called a Γn -contraction. Models for n-tuples of operators on Hilbert spaces of analytic functions on polydisc and Euclidean ball in Cn has been studied extensively. It is natural to look for models for n-tuples of operators associated with inhomogeneous domains [3, Corollary 4.4]. The class of Γn -contraction is perhaps the first such class to be investigated. Agler and Young [2] found that many of the fundamental results (e.g. substantial part of the Nagy–Foias model theory) in the theory of single contraction has close parallels for Γ2 -contractions; it is a natural idea to extend the model theory for general n ∈ N. In this paper we investigate properties of Γn -contractions and we give a model for Γn -isometries. As an application, we prove a Beurling–Lax–Halmos type theorem characterizing joint invariant subspaces of a pure Γn -isometry. Analogous result for the case n = 2 is done in [17]. We construct a large class of examples which provides some functional models for Γn -contractions on weighted Bergman spaces A(λ) (Dn ), for λ > 1 (and the limiting case H 2 (Dn ), Hardy space on polydisc). We show that these spaces, using basic representation theory of the symmetric group, split up into the orthogonal direct sum of subspaces indexed by p partitions of n. Further, each of these subspaces is invariant under multiplication by all the elementary symmetric functions and the n-tuple of operators (Ms1 , . . . , Msn ) restricted to any of these subspaces is a Γn -contraction. The question whether the aforementioned Γn -contractions are unitarily inequivalent for different choices of λ and p is intriguing and we will come back to this problem in a future work. Here, it is shown that the Γ2 -contractions on these spaces are unitarily inequivalent for different choices of λ and p. Although a Γ2 -contraction can be obtained by symmetrizing any pair of commuting contractions [2], it is no longer true that the symmetrization of any n-tuple of commuting contractions will necessarily give rise to a Γn -contraction, if n > 2 (see Remark 2.11). In fact, the symmetrization of an n-tuple of commuting contractions (T1 , . . . , Tn ) is a Γn -contraction if and only if (T1 , . . . , Tn ) satisfies the analogue of von Neumann’s inequality for all symmetric polynomials in n variables (see Proposition 2.12). However, it is shown in [2, Examples 1.7 and 2.3] that not all Γ2 -contractions are obtained in this way. We usually denote a typical point of Γn by (s1 , . . . , sn ). We shall also use the notation (S1 , . . . , Sn ) for an n-tuple of commuting operators associated in some way with Γn . In this paper an operator will always be a bounded linear operator on a Hilbert space. The polynomial ring in n variables over the field of complex numbers is denoted by

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C[z1 , . . . , zn ]. Consider a commuting n-tuple (S1 , . . . , Sn ) of operators. We say that Γn is a spectral set for (S1 , . . . , Sn ), or that (S1 , . . . , Sn ) is a Γn -contraction, if, for every polynomial p ∈ C[z1 , . . . , zn ],     p(S1 , . . . , Sn )  sup p(z) = p∞,Γ . n

(1.1)

z∈Γn

Furthermore, Γn is said to be a complete spectral set for (S1 , . . . , Sn ), or (S1 , . . . , Sn ) to be a complete Γn -contraction, if, for every matricial polynomial p in n variables,     p(S1 , . . . , Sn )  sup p(z). z∈Γn

Here, if S1 , . . . , Sn act on a Hilbert space H and the matricial polynomial p is given by p = [pij ] of order m × , where each pij is a scalar polynomial, then p(S1 , . . . , Sn ) denotes the operator from H to Hm with block matrix [pij (S1 , . . . , Sn )]. It is a deep result [2, Theorem 1.5] that a Γn -contraction is always a complete Γn -contraction and vice versa, for n = 2. It is not clear whether a similar result is true for n > 2. In case n = 2, (S1 , S2 ) is a Γ2 -contraction if and only if the operator ρ(αS1 , α2 S2 ) is positive for all α ∈ D [1, Theorem 1.5], where ρ is a hereditary polynomial defined by   ρ(S1 , S2 ) = 2 I − S2∗ S2 − S1 + S1∗ S2 − S2∗ + S2∗ S1 . For the case n > 2, this hereditary polynomial was generalized to ρn in [14]. However, it is proved in [14] that if Γn is a complete spectral set for a commuting n-tuple (S1 , . . . , Sn ), then ρn (αS1 , . . . , αn Sn ) is positive for all α ∈ D. We prove here that ρn (αS1 , . . . , αn Sn ) is positive for all α ∈ D even if Γn is just a spectral set for a commuting n-tuple (S1 , . . . , Sn ). The sufficiency part of this question is not known yet. We denote the unit circle by T. The distinguished boundary of Γn , denoted by bΓn , defined to be the Silov boundary of the algebra of functions which are continuous on Γn and analytic on the interior of Γn , is s(Tn ) [6, Lemma 8]. We shall use some spaces of vector-valued and operator-valued functions, we recall them following [2]. Let E be a separable Hilbert space. We denote by L(E) the space of operators on E, with the operator norm. Let H 2 (E) denote the usual Hardy space of analytic E-valued functions on D and L2 (E) the Hilbert space of square integrable E-valued functions on T, with their natural inner products. Let H ∞ L(E) denote the space of bounded analytic L(E)-valued functions on D and L∞ L(E) the space of bounded measurable L(E)-valued functions on T, each with appropriate version of the supremum norm. For ϕ ∈ L∞ L(E) we denote by Mϕ the Toeplitz operator with symbol ϕ, given by Mϕ f = P+ (ϕf ),

f ∈ H 2 (E),

where P+ : L2 (E) −→ H 2 (E) is the orthogonal projection. In particular Mz is the unilateral shift operator on H 2 (E) (the identity function on T will be denoted by z) and Mz¯ is the backward shift on H 2 (E).

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The symmetrization map s is a proper holomorphic map, Dn = s−1 (Γn ) = s−1 (s(Dn )) and Dn is polynomially convex. Therefore, s(Dn ) = Γn is polynomially convex by [19, Theorem 1.6.24]. Thus one notes that the Taylor spectrum of a Γn -contraction is contained in Γn . Although we have defined Γn -contractions by requiring that the inequality (1.1) holds for all polynomials p in C[z1 , . . . , zn ], this is equivalent to the definition of Γn -contractions by requiring (1.1) to hold for all functions p analytic in a neighbourhood of Γn due to polynomial convexity of Γn and the Oka–Weil theorem. As discussed in [2], the subtleties surrounding the various notions of joint spectrum and functional calculus for commuting tuples of operators are not relevant to this paper, simply because of the polynomial convexity of Γn . 2. Γn and Γn -contractions Note that (s1 , . . . , sn ) ∈ Γn if and only if all the zeros of the polynomial n n−i sn−i z i lie in D. This realization of points of Γn will be used repeatedly. We i=0 (−1) state two theorems about location of zeros of polynomials which will be useful in the sequel. For a polynomial p ∈ C[z], the derivative of p with respect to z will be denoted by p . Theorem 2.1. (See Gauss–Lucas, [12, p. 22].) The zeros of the derivative of a polynomial p lie in the convex hull of the zeros of p. We recall a definition. Definition 2.2. A polynomial p ∈ C[z] of degree d is called self-inversive if z d p( z1¯ ) = ωp(z) for some constant ω ∈ C with |ω| = 1. Theorem 2.3. (See Cohn, [12, p. 206].) A necessary and sufficient condition for all the zeros of a polynomial p to lie on the unit circle T is that p is self-inversive and all the zeros of p lie in the closed unit disc D. We shall need the following characterizations of the distinguished boundary of Γn . Theorem 2.4. Let si ∈ C, i = 1, . . . , n. The following are equivalent: (i) (s1 , . . . , sn ) is in the distinguished boundary of Γn ; (ii) |sn | = 1, s¯n si = s¯n−i and (γ1 s1 , . . . , γn−1 sn−1 ) ∈ Γn−1 , where γi = n−i n for i = 1, . . . , n − 1; (iii) (s1 , . . . , sn ) ∈ Γn and |sn | = 1; (iv) there exists (μ1 , . . . , μn−1 ) in the distinguished boundary of Γn−1 such that sj = μj + μ ¯n−j sn , j = 1, . . . , n − 1.

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Proof. Throughout this proof, we put s0 = 1. Let (s1 , . . . , sn ) be in the distinguished boundary of Γn . By definition, there are λi ∈ T, i = 1, . . . , n, such that si =



λk1 . . . λki

for i = 1, . . . , n.

(2.1)

1k1 <···
This is equivalent to the fact that the polynomial p, given by

p(z) =

n 

(−1)n−i sn−i z i

(2.2)

i=0

has all its zeros on T. Moreover, we clearly have |sn | = 1 and s¯n si = s¯n−i for i = 1, . . . , n − 1. It follows from Theorem 2.3 that the polynomial p (z) =

n 

(−1)n−i isn−i z i−1

(2.3)

i=1

has all its roots in D, which is equivalent to the fact that (γ1 s1 , . . . , γn−1 sn−1 ) ∈ Γn−1 , where γi = n−i n for i = 1, . . . , n − 1. Therefore (i) implies (ii). Conversely, considering the polynomial in Eq. (2.2), we observe that znp

  n 1 = (−1)i s¯i z i z¯ i=0

and (−1)n sn z n p

 1 = p(z) z¯

by the first part of (ii). Therefore p is a self-inversive polynomial. Note that all the roots of p lies in D as (γ1 s1 , . . . , γn−1 sn−1 ) ∈ Γn−1 , where γi = n−i n for i = 1, . . . , n − 1. Thus, it follows from Theorem 2.3 that p has all its roots on T. This is the same as saying that (s1 , . . . , sn ) is in the distinguished boundary of Γn . Clearly, (i) implies (iii). To see the converse, we note that (iii) implies that there exist λi ∈ D, i = 1, . . . , n, such that (2.1) holds and |sn | = |λ1 | · · · |λn | = 1. So λi ∈ T for all i = 1, . . . , n. It follows from the definition and (2.1) that (s1 , . . . , sn ) is in the distinguished boundary of Γn . To show (i) implies (iv), let (s1 , . . . , sn ) ∈ bΓn . Clearly, sn = λ1 . . . λn = eiθ for some ¯ n−1 . Since ¯1 . . . λ θ in R. So λn = eiθ λ sj =



λk1 · · · λkj

with |λj | = 1

1k1
for j = 1, . . . , n, we have sj =

 1k1
λk1 · · · λkj = μj + μ ¯j−1 λn

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where μj =



λk1 · · · λkj .

1k1
¯ n−1 we obtain ¯1 . . . λ Thus (μ1 , . . . , μn−1 ) ∈ bΓn−1 . Since λn = eiθ λ ¯ n−1 = μj + μj−1 μ ¯1 . . . λ sj = μj + μ ¯j−1 eiθ λ ¯n−1 eiθ = μj + μ ¯n−j eiθ , as μ ¯n−1 μj−1 = μ ¯n−j , j = 1, . . . , n − 1. Conversely, let μj = sj (λ) for some λ = ¯n−1 sn ∈ T. Thus sj = μj + μ ¯n−j λn μn−1 = (λ1 , . . . , λn−1 ) ∈ Tn−1 . We choose λn = μ μj + μj−1 λn = sj (λ1 , . . . , λn ) and sn = sn (λ1 , . . . , λn ) and hence (s1 , . . . , sn ) is in bΓn . Hence (iv) implies (i). 2 Lemma 2.5. If (s1 , . . . , sn ) ∈ Γn , then (γ1 s1 , . . . , γn−1 sn−1 ) ∈ Γn−1 , where γi = i = 1, . . . , n − 1.

n−i n

for

n Proof. Since (s1 , . . . , sn ) ∈ Γn , the polynomial p(z) = i=0 (−1)n−i sn−i z i , has all its roots in the closed unit disc D. It follows from Theorem 2.1 that the polynomial p (z) = n n−i isn−i z i−1 has all its roots in the closed unit disc as well. Hence we have i=1 (−1) the desired conclusion. 2 Remark 2.6. Considering the map π : Cn −→ Cn−1 defined by π(z1 , . . . , zn ) = (γ1 z1 , . . . , γn−1 zn−1 ) the above lemma can be restated as π(Γn ) ⊆ Γn−1 . If the map π were onto, then it would give a way to go back and forth between lower dimensional symmetrized polydisc to higher dimensional one. However, this is not true. For example, consider the quadratic polynomial z 2 − a2 = 0 where a ∈ (−1, 1). Clearly, (0, −a2 ) ∈ Γ2 . Our query then translates to whether there exists s3 ∈ C such that (0, −3a2 , s3 ) in Γ3 , that is, the cubic polynomial z 3 − 3a2 z + s3 has all its roots inside D. This is again equivalent to asking whether the cubic polynomial s¯3 z 3 − 3a2 z 2 + 1 has all its roots outside D. By the Schur–Cohn algorithm [9, Theorem 6.8b], this happens if and only if 1 − |s3 |2 and (1 − |s3 |2 ) − 9a4 are strictly positive. Thus a necessary condition is a4 < 19 . So if a2 is chosen to be in ( 13 , 1), for example a = 34 , the Schur–Cohn algorithm shows that the point (0, −a2 ) has no preimage under π. We have an analogue of the previous lemma for operators. Lemma 2.7. If (S1 , . . . , Sn ) is a Γn -contraction, then (γ1 S1 , . . . , γn−1 Sn−1 ) is a Γn−1 -contraction, where γi = n−i n for i = 1, . . . , n − 1.

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Proof. For p ∈ C[z1 , . . . zn−1 ], we note that p ◦ π ∈ C[z1 , . . . , zn ] and by hypothesis     p(γ1 S1 , . . . , γn−1 Sn−1 ) = p ◦ π(S1 , . . . , Sn )  p ◦ π∞,Γn = p∞,π(Γn )  p∞,Γn−1 . This completes the proof. 2 Lemma 2.8. If (s1 , . . . , sn ) ∈ Γn , then (α + s1 , αs1 + s2 , . . . , αsn−1 + sn , αsn ) ∈ Γn+1 for all α in D. Proof. If (s1 , . . . , sn ) ∈ Γn , then it follows from the definition of Γn that there are λk ∈ D, k = 1, . . . , n such that si =



λk1 · · · λki ,

i = 1, . . . , n.

1k1
If we let λn+1 = α, for some α ∈ D, then (˜ s1 , . . . , s˜n+1 ) ∈ Γn+1 , where s˜i =



λk1 · · · λki ,

i = 1, . . . , n + 1.

1k1
Putting s0 = 1 and sn+1 = 0, we note that s˜i = αsi−1 + si , i = 1, . . . , n + 1. Therefore we have the desired conclusion. 2 Remark 2.9. For any α ∈ C, considering the one-to-one map πα : Cn −→ Cn+1 defined by πα (z1 , . . . , zn ) = (α + z1 , αz1 + z2 , . . . , αzn−1 + zn , αzn ), the above lemma can be restated as πα (Γn ) ⊆ Γn+1 for all α ∈ D. Here is an operator analogue of the previous lemma. Lemma 2.10. Let (S1 , . . . , Sn ) be a Γn -contraction, then (α + S1 , αS1 + S2 , . . . , αSn−1 + Sn , αSn ) is a Γn+1 -contraction for all α in D. Proof. For p ∈ C[z1 , . . . , zn+1 ], we observe that p ◦ πα ∈ C[z1 , . . . , zn ], so by hypothesis we have   p(α + S1 , αS1 + S2 , . . . , αSn−1 + Sn , αSn )   = p ◦ πα (S1 , . . . , Sn )  p ◦ πα ∞,Γn

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= p∞,πα (Γn )  p∞,Γn+1 . Hence the proof. 2 For an n-tuple T = (T1 , . . . , Tn ) of commuting operators on a Hilbert space H, let s(T) := (s1 (T), . . . , sn (T)). We call s(T), the symmetrization of T. A polynomial p ∈ C[z1 , . . . , zn ] is called symmetric if p(σ.z) := p(zσ(1) , . . . , zσ(n) ) = p(z) for all z ∈ Cn and σ ∈ Sn , where Sn is the symmetric group on n symbols. If p ∈ C[z1 , . . . , zn ] is symmetric, then there is a unique q ∈ C[z1 , . . . , zn ] such that p = q ◦ s, where s is the symmetrization map [7, Theorem 3.3.1]. Remark 2.11. It is natural to ask if Lemma 2.10 remains valid when α is replaced by a contractive operator T commuting with all the Si ’s, i = 1, . . . , n. However, this is no longer true. We take n = 2 and give an example of a Γ2 -contraction (S1 , S2 ) and a contraction T such that (T + S1 , T S1 + S2 , T S2 ) is not a Γ3 -contraction. Let (A1 , A2 , A3 ) be a tuple of commuting contractions as in Kaijser–Varopoulos [15, Example 5.7]. We take S1 = A1 + A2 , S2 = A1 A2 and T = A3 . Clearly, (S1 , S2 ) = (A1 + A2 , A1 A2 ) is a Γ2 -contraction due to Ando’s inequality. But (T + S1 , T S1 + S2 , T S2 ) = s(A1 , A2 , A3 ) is not a Γ3 -contraction due to the failure of von Neumann’s inequality for more than two commuting contractions. Consider the symmetric polynomial p(z1 , z2 , z3 ) = z12 + z22 + z32 − 2z1 z2 − 2z2 z3 − 2z3 z1 in [15, Example 5.7]. Taking q to be the polynomial in 3-variables such that q ◦ s = p, where s is the symmetrization map, one observes that Γ3 cannot be a spectral set for s(A1 , A2 , A3 ). Proposition 2.12. The symmetrization of an n-tuple of commuting contractions (T1 , . . . , Tn ) is a Γn -contraction if and only if (T1 , . . . , Tn ) satisfies the analogue of von Neumann’s inequality for all symmetric polynomials in C[z1 , . . . , zn ]. Proof. Let T = (T1 , . . . , Tn ) be a commuting n-tuple of contractions satisfying the analogue of von Neumann’s inequality for all symmetric polynomials C[z1 , . . . , zn ]. We show that s(T) = (s1 (T), . . . , sn (T)) is a Γn -contraction. For a polynomial p ∈ C[z1 , . . . , zn ], we note that      p s1 (T), . . . , sn (T)  = p ◦ s(T)  p ◦ s

∞,Dn

= p∞,s(Dn ) ,

since p ◦ s is a symmetric polynomial, the above inequality holds by hypothesis, and it shows that Γn = s(Dn ) is a spectral set for (s1 (T), . . . , sn (T)).

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Conversely, let T = (T1 , . . . , Tn ) be a tuple of commuting contractions such that s(T) is a Γn -contraction and q ∈ C[z1 , . . . , zn ] be symmetric. So there is a p ∈ C[z1 , . . . , zn ] such that p ◦ s = q. By hypothesis, we have      q(T) = p s(T)   p∞,Γ = p ◦ s n ∞,Dn = q∞,Dn .

2

Next, we give a straightforward generalization of Lemma 3.2 in [4]. Proposition 2.13. Let T = (T1 , . . . , Tn ) be a commuting n-tuple of contractions on a Hilbert space H satisfying the analogue of von Neumann’s inequality for all symmetric polynomials in C[z1 , . . . , zn ]. Let M be an invariant subspace for si (T), i = 1, . . . , n. Then (s1 (T)|M , . . . , sn (T)|M ) is a Γn -contraction on the Hilbert space M. Proof. For a polynomial p ∈ C[z1 , . . . , zn ], we note that          p s1 (T) , . . . , sn (T)  = p s1 (T), . . . , sn (T)   M M M       p s1 (T), . . . , sn (T)  p∞,Γn , since s(T) = (s1 (T), . . . , sn (T)) is a Γn -contraction on H, the last inequality holds by the previous proposition. Hence we have the desired conclusion. 2 Remark 2.14. 1. It is clear from the proof of the proposition above that if (S1 , . . . , Sn ) is a Γn -contraction on a Hilbert space H and M is a common invariant subspace for Si , i = 1, . . . , n, then (S1 |M , . . . , Sn |M ) is a Γn -contraction on M. 2. As an immediate consequence of Proposition 2.12, we observe that the symmetrizations of the classes of n-tuples of commuting contractions discussed in [8] give rise to a large class of Γn -contractions. 3. It is shown in [4] that applying Lemma 3.2, a large class of Γ2 -contractions can be constructed. In an analogous way, examples of Γn -contractions can be constructed for any integer n > 2 applying Proposition 2.13. This procedure is illustrated in Section 5. 4. From Theorem 3.2 in [2], it is clear that all Γ2 -contractions are obtained by applying Lemma 3.2 in [4] as described in the same paper. However, for n  3, it is not clear whether all Γn -contractions have similar realizations. 2.1. A necessary condition for (S1 , . . . , Sn ) to be a Γn -contraction Fix x0 = y0 = 1. For α ∈ C, we define polynomials gα and hα in n-variables by the formulae

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gα (x) =

6233

n  (−1)i α ¯ i xi i=0

n  hα (x) = (−1)i αi xn−i i=0

where x = (x1 , . . . , xn ). Lemma 2.15. For α ∈ C, x = (x1 , . . . , xn ) and y = (y1 , . . . , yn ), the following holds: gα¯ (y)gα (x) − hα¯ (y)hα (x) =

n 



 (−1)i+j sgn n − (i + j) 1 − |α|2|n−(i+j)| Aα (i, j)|α|2B(i,j) yi xj

i,j=0

where Aα (i, j) =

αi−j α ¯ j−i

i  j; i < j,

B(i, j) = min{i, j, n − i, n − j}. Lemma 2.16. Let z ∈ Cn . Then z ∈ Γn if and only if gα (z) = 0 for all α ∈ D. Proof. If z = s(λ), by definition z ∈ Γn if and only if λ = (λ1 , . . . , λn ) ∈ Dn . Now

n gα (z) = i=1 (1 − α ¯ λi ) = 0 for all α ∈ D if and only if λi = α1¯ for all α ∈ D and i = 1, . . . , n, that is, λi ∈ D for i = 1, . . . , n. This completes the proof. 2 Lemma 2.17. For any commuting tuple (S1 , . . . , Sn ) of operators, σ(S1 , . . . , Sn ) ⊂ Γn if and only if gα (S1 , . . . , Sn ) is invertible for all α ∈ D. / Proof. By definition, for any α ∈ D, gα (S1 , . . . , Sn ) is invertible if and only if 0 ∈ σ(gα (S1 , . . . , Sn )). From spectral mapping theorem, it follows that 0 ∈ / gα (σ(S1 , . . . , Sn )). In other words, gα does not vanish anywhere on σ(S1 , . . . , Sn ). Equivalently, by Lemma 2.16, σ(S1 , . . . , Sn ) ⊂ Γn . 2 For an n-tuple of commuting operators (S1 , . . . , Sn ) on the Hilbert space H, define the hereditary polynomial ρn (S1 , . . . , Sn ) ∈ L(H) by the formula ρn (S1 , . . . , Sn ) =

n 

 (−1)i+j n − (i + j) Si∗ Sj

(2.4)

i,j=0

In the following proposition, we obtain a necessary condition for a commuting tuple (S1 , . . . , Sn ) to be a Γn -contraction in terms of positivity of the Hermitian operator pencil ρn (βS1 , . . . , β n Sn ).

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Proposition 2.18. Let (S1 . . . , Sn ) be a Γn -contraction, then   ρn βS1 , . . . , β n Sn  0

for all β ∈ D.

Proof. Let hα (z)  λi − α = , gα (z) 1−α ¯ λi i=1 n

ψα (z) :=

(2.5)

where z = s(λ) for λ = (λ1 , . . . , λn ) ∈ Dn . Since the pull back of the map ψα by the symmetrization map is an inner function on Dn and analytic on a neighbourhood of Dn , the map ψα is an inner function on Γn which is analytic on a neighbourhood of Γn whenever α ∈ D. We further note that ψα (S1 , . . . , Sn ) is a contraction for all α ∈ D as Γn is a spectral set for the commuting tuple (S1 . . . , Sn ) and ψα has supremum norm on Γn equal to 1. For any β ∈ T,   β n hαβ¯ (βz1 , . . . , β n zn ) . ψα βz1 , . . . , β n zn = gαβ¯ (βz1 , . . . , β n zn ) Thus we have ψα (βS1 , . . . , β n Sn ) = β n ψαβ¯ (S1 , . . . , Sn ), and therefore,    ψα βS1 , . . . , β n Sn   1,

(2.6)

for all α ∈ D and β ∈ T. Since the mapping β → ψα (βS1 , . . . , β n Sn ) is analytic on a neighbourhood of the closed unit disc, it follows from the maximum modulus principle that the inequality (2.6) holds for all α, β ∈ D, and hence  ∗   1 − ψα βS1 , . . . , β n Sn ψα βS1 , . . . , β n Sn  0, for all α, β ∈ D. From the definition of ψα , we deduce that  ∗    ∗   gα βS1 , . . . , β n Sn gα βS1 , . . . , β n Sn − hα βS1 , . . . , β n Sn hα βS1 , . . . , β n Sn  0, for all α, β ∈ D. Therefore, by Lemma 2.15 

1 − |α|

2

n  i,j=0

i+j

(−1)

 |n−(i+j)|−1  

 2k sgn n − (i + j) |α| Aα (i, j)|α|2B(i,j) β¯i β j Si∗ Sj  0 k=0

for all α, β ∈ D. Clearly, the expression above remains positive without 1 − |α|2 . Now letting α tend to 1, we have   ρn βS1 , . . . , β n Sn  0.

2

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3. Some functional models of Γn -contractions on weighted Bergman spaces In this section, we describe a class of reproducing kernel Hilbert spaces and tuples of naturally occurring operators on them and show that they are Γn -contractions. In order to do this, we recall some notations and terminologies. For λ > 1, let

n n 2 λ−2 dV (λ) := ( λ−1 ri dri dθi ) be a measure on the polydisc Dn . The π ) ( i=1 (1 − ri ) (λ) n weighted Bergman space A (D ) on the polydisc Dn is the subspace of the Hilbert space L2 (Dn , dV (λ) ) consisting of holomorphic functions. The reproducing kernel K of A(λ) (Dn ) is given by K(z, w) =

n 

(1 − zi w ¯i )−λ ,

z = (z1 , . . . , zn ), w = (w1 , . . . , wn ) ∈ Dn .

i=1

The limiting case of λ = 1, as is well-known, is the Hardy space on the polydisc. In the following discussion we shall talk about the weighted Bergman spaces A(λ) (Dn ) for λ  1. Let Sn denote the permutation group of n symbols. The natural action of Sn on Cn is given as follows: For (σ, z) ∈ Sn × Cn , (σ, z) → σ.z := (zσ(1) , . . . , zσ(n) ). A finite sequence p = (p1 , . . . , pk ) of positive integers with p1  · · ·  pk is a partition of n, k  if i=1 pi = n, denoted by p n. The set Sn of equivalence classes of irreducible representations (hence finite dimensional) of the permutation group Sn is parametrized by the partitions p of n. The immanant of the matrix ((aij ))ni,j=1 corresponding to the partition p of n is denoted by imm(p) ((aij ))ni,j=1 and is defined by the equation [11, p. 81] imm(p) ((aij ))ni,j=1 =

 σ∈Sn

χp (σ)

n 

aiσ(i) ,

(3.1)

i=1

where χp is the character of the representation corresponding to the partition p of n. The permanent and the determinant of a matrix are special cases of immanants. For permanent, note that the character χ(n) is identically 1 on Sn , where (n) denotes the partition (n, 0, . . . , 0) that corresponds to the trivial representation. For determinant, the character χ(1n ) is 1 for an even permutation and −1 for an odd permutation in Sn , where (1n ) denotes the partition (1, . . . , 1) that corresponds to the sign representation. Now we are ready to describe the reproducing kernel Hilbert spaces and tuples of naturally occurring multiplication operators on them. Define for a partition p of n, the linear map Pp : A(λ) (Dn ) −→ A(λ) (Dn ) by (Pp f )(z) =

χp (1)  χp (σ)f (σ.z), n!

(3.2)

σ∈Sn

where χp is the character of the representation corresponding to the partition p of n. Lemma 3.1. For partitions p and q of n, one has Pp Pq = δpq Pp .

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Proof.  (Pp Pq f )(z) = Pp

χq (1)  χq (σ)f (σ.z) n! σ∈Sn

χq (1)  χq (σ)(Pp f )(σ.z) = n! σ∈Sn

χq (1)  χp (1)  χq (σ) χp (τ )f (τ σ.z) n! n! σ∈Sn τ ∈Sn   χp (1)χq (1)  χq (σ)χp (τ ) f (ν.z) = (n!)2 ν∈Sn τ σ=ν    −1  χp (1)χq (1)  χ (σ)χ νσ f (ν.z) = q p (n!)2 =

ν∈Sn

σ∈Sn

 Since χp ∗ χq (ν) = σ∈Sn χq (σ)χp (νσ −1 ) = δpq χpn!(1) χp (ν) [18, Theorem III.2.7.], the conclusion follows. 2 Lemma 3.2. The operator Pp is self-adjoint for any partition p of n. Proof. Since the monomials z I , I = (i1 , . . . , il , . . . , in ), il ∈ N ∪ {0} form an orthogonal basis, it is enough to prove that 

   P p z I , z J = z I , Pp z J

(3.3)

for every pair of multi-indices I and J. Now  χp (1)    Pp z I , z J = χp (σ) z I (σ.z), z J n!



σ∈Sn

i

where z I (σ.z) = z1σ

−1 (1)

i

. . . znσ



−1 (n)

 χp (1) Pp z I , z J = n! =

. Thus 

 i −1 2 i −1 χp (σ)z1σ (1) . . . znσ (n) 

σ∈Sn :il =jσ(l)

 χp (1)  z I 2 n!



χp (σ)

σ∈Sn :il =jσ(l)

n! as z I 2 = (λ)ii1 !...i = z I (σ.z)2 = z σ.I 2 for any permutation σ, where (λ)il = 1 ...(λ)in λ . . . (λ + il − 1). Similarly

 χp (1)    χp (1) I 2 z  z I , Pp z J = χp (σ) z I , z J (σ.z) = n! n!



σ∈Sn

 σ∈Sn :il =jσ(l)

  χp σ −1 .

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Since χp (σ −1 ) = χp (σ) for any σ ∈ Sn , comparing the equations above, it follows that (3.3) holds. This completes the proof. 2 Combining Lemma 3.1 and Lemma 3.2, we have the following corollary. Corollary 3.3. The operator Pp is an orthogonal projection for any partition p of n. One also observes that 

(Pp f )(z) =

pn

 χp (1)  pn

n!

χp (σ)f (σ.z)

σ∈Sn

 1   χp (1)χp (σ) f (σ.z) = f (z), = n! σ∈Sn

pn

where the last equality follows from Schur orthogonality relations [18, Theorem III.2.8]. Thus we have the following lemma. Lemma 3.4.

 pn

Pp = I.

For the sake of completeness, we include the following elementary lemma. Lemma 3.5. If A = ((aij ))ni,j=1 is a square matrix of order n, then imm(p) Atr = imm(p) A for any partition p of n, where Atr denotes the transpose of the matrix A. Proof. We have imm

(p)

tr

A =



χp (σ)

aσ(i)i =

i=1

σ∈Sn

=

n 

 σ∈Sn



n   χp σ −1 aiσ−1 (i) ,

σ∈Sn

i=1

χp (σ)

n 

aiσ−1 (i)

i=1

where the last equality holds since χp (σ −1 ) = χp (σ), for every partition p of n and σ ∈ Sn . Now imm

(p)

tr

A =

 σ∈Sn

χp (σ)

n 

aiσ(i) = imm(p) A,

i=1

where the last equality follows from the fact that χp (σ) is an integer for every partition p of n and σ ∈ Sn [10, Corollary 22.17]. 2 Clearly, the subspace Pp (A(λ) (Dn )) of A(λ) (Dn ) is a reproducing kernel Hilbert space for any partition p of n. In the next proposition we give a formula for the reproducing (λ) kernel Kp of the subspace Pp (A(λ) (Dn )).

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(λ)

Proposition 3.6. The reproducing kernel Kp Kp(λ) (z, w) =

is given explicitly by the formula:

 n χp (1) imm(p) (1 − zi w ¯j )−λ i,j=1 , n!

z, w ∈ Dn ,

where p is a partition of n.

n ¯i )−λ . Proof. The reproducing kernel for A(λ) (Dn ) is given by K (λ) (z, w) = i=1 (1 − zi w (λ) (λ) n (λ) Therefore the reproducing kernel Kp for the subspace Pp (A (D )) of A (Dn ) is given by   Kp(λ) (z, w) = Kp(λ) w (z)   (λ) (z) = P p Kw χp (1)  (λ) = χp (σ)Kw (σ.z) n! σ∈Sn

=

n  χp (1)  χp (σ) (1 − zσ(i) w ¯i )−λ n! i=1 σ∈Sn

=

 n χp (1) imm(p) (1 − zi w ¯j )−λ i,j=1 , n!

where the last equality follows from Lemma 3.5. 2 Remark 3.7. In [13], for the partition (1n ), an orthonormal basis for the space P(1n ) (A(λ) (Dn )) consisting of anti-symmetric functions is obtained explicitly and thereby the reproducing kernel (λ)

K(1n ) (z, w) =

 n 1 det (1 − zi w ¯j )−λ i,j=1 , n!

z, w ∈ Dn ,

is computed. We recall that sk , k = 1, . . . , n is the elementary symmetric function of degree k in n variables. Let Msk denote the multiplication by sk , k = 1, . . . , n, on A(λ) (Dn ). We have the following proposition. Proposition 3.8. The subspace Pp (A(λ) (Dn )) is invariant under Msk for any partition p of n and k = 1, . . . , n. In other words, Pp Msk = Msk Pp for any partition p of n and k = 1, . . . , n. Proof. Let g ∈ Pp (A(λ) (Dn )), it is enough to show that sk g ∈ Pp (A(λ) (Dn )) for k = 1, . . . , n. Since g ∈ Pp (A(λ) (Dn )), Pp f = g for some f ∈ A(λ) (Dn ). Since sk is symmetric sk (σ.z) = sk (z) for all σ ∈ Sn , therefore

S. Biswas, S. Shyam Roy / Journal of Functional Analysis 266 (2014) 6224–6255

(sk g)(z) = sk (z)

6239

χp (1)  χp (σ)f (σ.z) n! σ∈Sn

=

χp (1)  χp (σ)sk (σ.z)f (σ.z) n! σ∈Sn

=

χp (1)  χp (σ)(sk f )(σ.z) n! σ∈Sn

= Pp (sk f )(z). Hence sk g ∈ Pp (A(λ) (Dn )). This completes the proof. 2 Remark 3.9. The previous proposition can be restated by saying that Pp (A(λ) (Dn )) is a joint reducing subspace for Msk , k = 1, . . . , n for every partition p of n. We consider the restriction of commuting n-tuple of multiplication operators (Ms1 , . . . , Msn ) to Pp (A(λ) (Dn )). For a measure space (X, μ) with measure μ and ϕ ∈ L∞ (X, μ), let Mϕ denote the operator on L2 (X, μ) defined by Mϕ f = ϕf for f ∈ L2 (X, μ). Let M := (Mz1 , . . . , Mzn ) denote the commuting tuple of multiplication operators by the coordinate functions on A(λ) (Dn ). We state and prove the following fact for the sake of completeness. Lemma 3.10. For λ  1, one has   p(M)  p ∞,Dn

for every p ∈ C[z1 , . . . , zn ].

In other words, the commuting tuple M acting on A(λ) (Dn ) satisfies the von Neumann inequality for λ  1. Proof. Letting .2 denote the L2 -norm on A(λ) (Dn ), λ  1, for f ∈ A(λ) (Dn ), we have   p(M)f  = Mp f 2  p ∞,Dn f 2 . 2 Hence the desired inequality. 2 As an immediate consequence of the previous lemma and Proposition 2.12, we have the following corollary. Corollary 3.11. The symmetrization of M, that is, s(M) = (Ms1 , . . . , Msn ) acting on A(λ) (Dn ) is a Γn -contraction. We now state the main result of this section which follows immediately from Proposition 3.8, Proposition 2.13 and Lemma 3.10. This exhibits a large class of reproducing kernel Hilbert spaces on each of which the naturally occurring tuple of multiplication operators is a Γn -contraction.

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Theorem 3.12. The commuting n-tuple (Ms1 , . . . , Msn ) of multiplication operators acting on Pp (A(λ) (Dn )) is a Γn -contraction, for every partition p of n and all λ  1. It is an interesting question whether the Γn -contractions described above are unitarily inequivalent for different choices of λ and p. We intend to discuss this problem in a future work. However, here we show that for the case n = 2, it is so for different choices of λ and p. Consider the weighted Bergman space A(λ) (D2 ) on the bidisc for λ  1 (for λ = 1, this discussion includes the Hardy space H 2 (D2 ) of bidisc) and its subspaces P(2) (A(λ) (D2 )) and P(12 ) (A(λ) (D2 )). Clearly,      2

  P(2) A(λ) D2 = A(λ) := f ∈ A(λ) D2 : f (z1 , z2 ) = f (z2 , z1 ) sym D

 = span z1m z2n + z1n z2m : m  n  0, (z1 , z2 ) ∈ D2 and     

  (λ)  P(12 ) A(λ) D2 = Aanti D2 := f ∈ A(λ) D2 : f (z1 , z2 ) = −f (z2 , z1 ) ,

 = span z1m z2n − z1n z2m : m > n  0, (z1 , z2 ) ∈ D2 . (λ)

(λ)

Note that Asym (D2 ) and Aanti (D2 ) denote the subspaces of symmetric functions and (λ) anti-symmetric functions and one has the orthogonal direct sum A(λ) (D2 ) = Asym (D2 ) ⊕ (λ) 2 Aanti (D ). Let G2 be the interior of Γ2 and A(G2 ) be the function algebra obtained by taking completion of the functions holomorphic in a neighbourhood of G2 with respect (λ) (λ) to supremum norm. It was shown in [13, p. 2363] that both Asym (D2 ) and Aanti (D2 ) are (λ) modules over A(G2 ) under the action (h, f ) → (h ◦ s)f , h ∈ A(G2 ), f ∈ Asym (D2 ) (or in (λ) Aanti (D2 )). (λ) (λ) For the sake of simplicity, we will denote the reproducing kernel of Asym (D2 ) by Ksym (λ) instead of K(2) which is given by    1 per (1 − zj w ¯k )−λ 2j,k=1 , 2 1 1 ¯1 )−λ (1 − z2 w ¯2 )−λ + (1 − z1 w ¯2 )−λ (1 − z2 w ¯1 )−λ = (1 − z1 w 2 2

(λ) Ksym (z, w) =

(λ)

(3.4)

(λ)

for z, w ∈ D2 . Similarly, we will denote the reproducing kernel of Aanti (D2 ) by Kanti (λ) instead of K(12 ) which is given by (λ)

   1 det (1 − zj w ¯k )−λ 2j,k=1 , 2 1 1 ¯1 )−λ (1 − z2 w ¯2 )−λ − (1 − z1 w ¯2 )−λ (1 − z2 w ¯1 )−λ = (1 − z1 w 2 2

Kanti (z, w) =

for z, w ∈ D2 .

(3.5)

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(λ)

The Hilbert space Asym (D2 ) can be thought of as a space of functions defined on the symmetrized bidisc G2 . We introduce the following notation:

  H(λ) (G2 ) := f : G2 −→ C is holomorphic: f ◦ s ∈ A(λ) D2 .

(3.6)

(λ)

Clearly, Asym (D2 ) and H(λ) (G2 ) are the same as sets. The inner product on H(λ) (G2 ) is given by f, gH(λ) (G2 ) := f ◦ s, g ◦ sA(λ) (D2 ) .

(3.7)

(λ)

The map from H(λ) (G2 ) to Asym (D2 ) defined by f → f ◦ s is a unitary which intertwines the pair (Mu1 , Mu2 ) with the pair (Mz1 +z2 , Mz1 z2 ), where u1 = z1 + z2 , u2 = z1 z2 . Thus (λ) H(λ) (G2 ) and Asym (D2 ) are unitarily equivalent as modules over A(G2 ). Therefore the (λ) reproducing kernel KG2 for H(λ) (G2 ) is given by  1 1 (λ)  KG2 s(z), s(w) = (1 − z1 w ¯1 )−λ (1 − z2 w ¯2 )−λ + (1 − z1 w ¯2 )−λ (1 − z2 w ¯1 )−λ 2 2  λ

(z1 + z2 )(w =1+ ¯1 + w ¯2 ) − 2z1 z2 w ¯1 w ¯2 2 2 λ(λ + 1) 

(z1 + z2 )(w + ¯1 + w ¯2 ) − 2z1 z2 w ¯1 w ¯2 8  + (z1 − z2 )2 (w ¯1 − w ¯2 )2 + higher order terms (λ)

for z, w ∈ D2 . Note that KG2 (s(z), 0) = 1 for z ∈ D2 . In terms of coordinates of symmetrized bidisc, one can write (λ)

λ (u1 v¯1 − 2u2 v¯2 ) 2    λ(λ + 1)

+ (u1 v¯1 − 2u2 v¯2 )2 + u21 − 4u2 v¯12 − 4¯ v2 8 + higher order terms

KG2 (u, v) = 1 +

(3.8)

for u = (u1 , u2 ), v = (v1 , v2 ) ∈ G2 . (λ) From [13, p. 2363], it follows that Aanti (D2 ) and the Bergman module A(λ) (G2 ) are (λ) unitarily equivalent as modules over A(G2 ). The reproducing kernel BG2 for A(λ) (G2 ) is given by  Js 2λ (1 − z1 w ¯1 )−λ (1 − z2 w ¯2 )−λ − (1 − z1 w ¯2 )−λ (1 − z2 w ¯1 )−λ (λ)  BG2 s(z), s(w) = 2 aδ (z)aδ (w) =1+

 λ + 1

(z1 + z2 )(w ¯1 + w ¯2 ) − 2z1 z2 w ¯1 w ¯2 2

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 2 (λ + 1)(λ + 2) 3

(z1 + z2 )(w ¯1 + w ¯2 ) − 2z1 z2 w ¯1 w ¯2 6 4  1 + (z1 − z2 )2 (w ¯1 − w ¯2 )2 4 +

+ higher order terms for z, w ∈ D2 , where Js (z) = aδ (z) = z1 − z2 and Js 2λ = λ2 [13, Theorem 2.3]. Note (λ) that BG2 (s(z), 0) = 1 for z ∈ D2 . In terms of coordinates of symmetrized bidisc, one can write λ+1 (u1 v¯1 − 2u2 v¯2 ) 2   2  (λ + 1)(λ + 2) 3 1 2 2 + (u1 v¯1 − 2u2 v¯2 ) + u1 − 4u2 v¯1 − 4¯ v2 6 4 4

(λ)

BG2 (u, v) = 1 +

+ higher order terms

(3.9)

for u = (u1 , u2 ), v = (v1 , v2 ) ∈ G2 . (λ)

(μ)

Proposition 3.13. The Hilbert modules Asym (D2 ) and Aanti (D2 ) over A(G2 ) are not unitarily equivalent for any λ, μ  1. (λ)

(μ)

Proof. Since Asym (D2 ) and H(λ) (G2 ) are unitarily equivalent and Aanti (D2 ) and A(μ) (G2 ) are unitarily equivalent (as modules over A(G2 )), it is enough to show that the commuting pair (Mu1 , Mu2 ) of multiplication operators by the coordinate functions on H(λ) (G2 ) and A(μ) (G2 ) are not unitarily equivalent. If possible, let they be unitarily equivalent. (λ) (λ) (λ) (μ) Both KG2 and BG2 are normalized kernels, that is, KG2 (u, 0) = BG2 (u, 0) = 1 for u ∈ G2 . Since by construction, the polynomial ring C[u1 , u2 ] in two variables is dense in both H(λ) (G2 ) and A(μ) (G2 ), therefore by [5, Lemma 4.8(c)], we have (λ)

(μ)

KG2 (u, v) = BG2 (u, v) for u, v ∈ G2 . Thus from Eqs. (3.8) and (3.9), it follows that μ+1 λ (λ) (μ) = ∂1 ∂¯1 KG2 (0, 0) = ∂1 ∂¯1 BG2 (0, 0) = 2 2 and 4

λ(λ + 1) (μ + 1)(μ + 2) (λ) (μ) = ∂12 ∂¯12 KG2 (0, 0) = ∂12 ∂¯12 BG2 (0, 0) = 4 4 6

which is a contradiction as λ = μ + 1 gives 4 = 6. This completes the proof. 2

(3.10)

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We specifically mention the case of Hardy space in particular, that is, when λ = μ = 1. 2 2 (D2 ) and Hanti (D2 ) over A(G2 ) are not uniProposition 3.14. The Hilbert modules Hsym tarily equivalent.

4. Γn -unitaries and Γn -isometries We start with the following obvious generalizations of definitions in [2]. Definition 4.1. Let S1 , . . . , Sn be commuting operators on a Hilbert space H. We say that (S1 , . . . , Sn ) is (i) a Γn -unitary if Si , i = 1, . . . , n are normal operators and the joint spectrum σ(S1 , . . . , Sn ) of (S1 , . . . , Sn ) is contained in the distinguished boundary of Γn ; (ii) a Γn -isometry if there exist a Hilbert space K containing H and a Γn -unitary (S˜1 , . . . , S˜n ) on K such that H is invariant for S˜i , i = 1, . . . , n and Si = S˜i |H , i = 1, . . . , n; (iii) a Γn -co-isometry if (S1∗ , . . . , Sn∗ ) is a Γn -isometry; (iv) a pure Γn -isometry if (S1 , . . . , Sn ) is a Γn -isometry and Sn is a pure isometry. The proof of the following theorem works along the lines of Agler and Young [2]. Theorem 4.2. Let Si , i = 1, . . . , n, be commuting operators on a Hilbert space H. The following are equivalent: (i) (S1 , . . . , Sn ) is a Γn -unitary; ∗ (ii) Sn∗ Sn = I = Sn Sn∗ , Sn∗ Si = Sn−i and (γ1 S1 , . . . , γn−1 Sn−1 ) is a Γn−1 -contraction, n−i where γi = n for i = 1, . . . , n − 1; (iii) there exist commuting unitary operators Ui for i = 1, . . . , n on H such that 

Si =

Uk1 . . . Uki

for i = 1, . . . , n.

1k1 <···
Proof. Suppose (i) holds. Let (S1 , . . . , Sn ) be a Γn -unitary. By the spectral theorem for commuting normal operators, there exists a spectral measure M (·) on σ(S1 , . . . , Sn ) such that  Si =

si (z)M (dz),

i = 1, . . . , n,

σ(S1 ,...,Sn )

where s1 , . . . , sn are the coordinate functions on Cn . Let τ be a measurable right inverse of the restriction of s to Tn , so that τ maps distinguished boundary of Γn to Tn . Let

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τ = (τ1 , . . . , τn ) and  Ui =

τi (z)M (dz),

i = 1, . . . , n.

σ(S1 ,...,Sn )

Clearly U1 , . . . , Un are commuting unitaries on H and    Uk1 . . . Uki = 1k1 <···
1k1 <···
 =

τk1 (z) . . . τki (z)M (dz)

σ(S1 ,...,Sn )

si (z)M (dz) σ(S1 ,...,Sn )

= Si , for i = 1, . . . , n. Hence (i) implies (iii). ∗ Suppose (iii) holds. Then Sn∗ Sn = I = Sn Sn∗ and Sn∗ Si = Sn−i , i = 1, . . . , n follow immediately. Moreover, since (S1 , . . . , Sn ) is a Γn -contraction, we have from Lemma 2.7 that (γ1 S1 , . . . , γn−1 Sn−1 ) is a Γn−1 -contraction, where γi = n−i n for i = 1, . . . , n − 1. Hence (iii) implies (ii). ∗ Suppose (ii) holds. Since Sn is normal, by Fuglede’s theorem Sn−i Sn−i = Sn∗ Si Si∗ Sn = ∗ ∗ ∗ ∗ Si Si . Now as Si ’s commute, Sn−i Sn−i = Sn Si Sn−i = Sn Sn−i Si = Si∗ Si and we have each of Si , i = 1, . . . , n is normal. So the unital C ∗ -algebra C ∗ (S1 , . . . , Sn ) generated by S1 , . . . , Sn is commutative and by Gelfand–Naimark’s theorem is ∗-isometrically isomorphic to C(σ(S1 , . . . , Sn )). Let Sˆ1 , . . . , Sˆn be the images of S1 , . . . , Sn under the Gelfand map. By definition, for an arbitrary point z = (s1 , . . . , sn ) in σ(S1 , . . . , Sn ), Sˆi (z) = si for i = 1, . . . , n. By properties of the Gelfand map and hypothesis we have, Sˆn (z)Sˆn (z) = 1 = Sˆn (z)Sˆn (z)

and Sˆn (z)Sˆi (z) = Sˆn−i (z)

for z in σ(S1 , . . . , Sn ).

Thus we obtain |sn | = 1, s¯n si = s¯n−i . Now (γ1 S1 , . . . , γn−1 Sn−1 ) is a Γn−1 -contraction implies   p(γ1 S1 , . . . , γn−1 Sn−1 )  p∞,Γ n−1 which is equivalent to p2∞,Γn−1 − p(γ1 S1 , . . . , γn−1 Sn−1 )∗ p(γ1 S1 , . . . , γn−1 Sn−1 ) is positive. Applying Gelfand transform we have  ∗   p2∞,Γn−1 − p γ1 Sˆ1 (z), . . . , γn−1 Sˆn−1 (z) p γ1 Sˆ1 (z), . . . , γn−1 Sˆn−1 (z)  0 for z in σ(S1 , . . . , Sn ). This shows (γ1 s1 , . . . , γn−1 sn−1 ) is in the polynomially convex hull of Γn−1 . Since Γn−1 is polynomially convex, (γ1 s1 , . . . , γn−1 sn−1 ) is in Γn−1 . Therefore by Theorem 2.4 (s1 , . . . , sn ) is in the distinguished boundary of Γn and hence σ(S1 , . . . , Sn ) ⊂ bΓn . This proves (ii) implies (i). 2

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It is easy to see that Lemma 3.1, Lemma 3.2, Corollary 3.3, Lemma 3.4, Proposition 3.8 and Lemma 3.10 hold with identical proofs for L2 (Tn ). The following theorem gives an interesting example of a Γn -unitary. Theorem 4.3. The commuting n-tuple (Ms1 , . . . , Msn ) of multiplication operators acting on Pp (L2 (Tn )) is a Γn -unitary, for every partition p of n. Proof. The commuting n-tuple (Ms1 , . . . , Msn ) of multiplication operators acting on L2 (Tn ) is a Γn -unitary by Theorem 4.2. For each partition p of n, Pp (L2 (Tn )) is a joint reducing subspace for Msk , k = 1, . . . , n, by Proposition 3.8. The desired conclusion follows from (ii) of Theorem 4.2 and Lemma 2.7. 2 Remark 4.4. It is of independent interest to explicitly describe the Γn -unitary (Ms1 , . . . , Msn ) as the symmetrization of a commuting n-tuple (U1 , . . . , Un ) of unitary operators on Pp (L2 (Tn )) as in Theorem 4.2. For the case n = 2, it is easy to write explicitly the unitaries for (Ms1 , Ms2 ) on L2sym (T2 ) as follows: M s 1 = Mf 1 + M f 2

and Ms2 = Mf1 f2

where f1 and f2 are symmetric L2 functions defined by f1 (z1 , z2 ) =

z1 ,

Re z1 > Re z2 ;

z2 ,

Re z1  Re z2 ,

and f2 (z1 , z2 ) =

z2 , Re z1 > Re z2 ; z1 , Re z1  Re z2 .

Clearly Mf1 and Mf2 are unitaries on L2sym (T2 ). One could construct the unitaries similarly on other pieces and in higher dimensions which is slightly more involved. One observes that similar result fails for Γ2 -isometries as shown in [2, Example 2.3] where one won’t be able to find holomorphic functions on all of D2 . It is not clear a priori whether unitarity of Sn implies that the Γn -contraction (S1 , . . . , Sn ) is a Γn -unitary. However, the following is true. Lemma 4.5. Let T = (T1 , . . . , Tn ) be a tuple of commuting contractions on a Hilbert

n space H. If sn (T)(= i=1 Ti ) is a unitary, then s(T) is a Γn -unitary. Proof. Note that each of the Ti , i = 1, . . . , n is invertible as they are commuting with each other and their product is unitary. Since each Ti is a contraction, we have Ti−1   1. First we show that Ti−1  = 1 for all i = 1, . . . , n. If not, then for some k in {1, . . . , n} we

n have Tk−1  > 1. So, there exists a y in H such that Tk−1 y > y. Let ( i=k Ti )y = x. Thus −1 −1   sn (T)−1   sn (T) x = Tk y · ny >1 x y ( i=k Ti )y

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−1

is a contradiction as by hypothesis sn (T) is also a unitary. Next we show that each Ti is isometry, that is, Ti∗ Ti = I, i = 1, . . . , n. Suppose for some  in {1, . . . , n}, T∗ T = I. There exists nonzero y in H such that (I − T∗ T )y = 0. Since T is a contraction, 1 y2 − T y2 = (I − T∗ T )y, y = (I − T∗ T ) 2 y2 > 0. But this shows, T−1  > 1 which is a contradiction. Thus Ti∗ Ti = I, i = 1, . . . , n. Applying the same trick to T∗ = (T1∗ , . . . , Tn∗ ), we see that each Ti is a unitary. Hence by part (iii) of the theorem above, s(T) is a Γn -unitary. 2 Remark 4.6. Let (S1 , . . . , Sn ) be a Γn -contraction. From Proposition 2.18, it follows that   ρn βS1 , . . . , β n Sn  0

for all β ∈ D.

Clearly this inequality remains valid for β ∈ T as well. From Eq. (2.4), for β ∈ T, one could write 

2

n

n−k



ρn βS1 , β S2 , . . . , β Sn =

n−1 2 ]  [

   ∗ (−1)k n − (2i + k) β k Si∗ Si+k − Sn−k−i Sn−i

k=1 i=0 n−k

+

n−1 2 ]  [

  ∗  ∗ (−1)k n − (2i + k) β¯k Si+k Si − Sn−i Sn−k−i

k=1 i=0 n

+

[2] 

  ∗ (n − 2i) Si∗ Si − Sn−i Sn−i

(4.1)

i=0

where [x] denotes the largest integer less than or equal to x. Along with Sn being unitary, if we assume further that n

[2] 

  ∗ (n − 2i) Si∗ Si − Sn−i Sn−i  0,

i=1

then we could say (S1 , . . . , Sn ) is a Γn -unitary. Integrating ρn (βS1 , β 2 S2 , . . . , β n Sn ) over T and using the assumption, we get n

[2] 

  ∗ (n − 2i) Si∗ Si − Sn−i Sn−i = 0.

(4.2)

i=1

We will prove the following generalization of Lemma 2.4 from [4]. Lemma 4.7. Let X1 , . . . , Xn be bounded operators on a Hilbert space H such that   Re βX1 + β 2 X2 + · · · + β n Xn  0 for all β ∈ T. Then Xi = 0 for each i = 1, . . . , n.

(4.3)

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From the previous lemma, it follows that [ n−k 2 ]



n − (2i + k)



 ∗ ∗ Si+k Si − Sn−i Sn−k−i = 0

(4.4)

i=0 ∗ for each k = 1, . . . , n − 1. Now consider Eq. (4.4) for k = n − 1, we have Sn−1 = Sn∗ S1 . ∗ ∗ Let k = n − i gives Sn−i = Sn Si for all i < 2m. For i = 2m, Eq. (4.4) gives m 

 ∗  ∗ 2(m − i) Si+n−2m Si − Sn−i S2m−i = 0.

(4.5)

i=0 ∗ But for 0 < i < 2m, by induction hypothesis we have Si+n−2m Si = Sn∗ S2m−i Si = ∗ ∗ ∗ Sn−i S2m−i . Thus Eq. (4.5) reduces to Sn S2m = Sn−2m . Similarly, using Eq. (4.5) and ∗ ∗ . Thus Sn−i = Sn∗ Si for all the induction hypothesis, we have Sn∗ S2m+1 = Sn−(2m+1) i  n − 1. This shows that (S1 , . . . , Sn ) is a Γn -unitary.

Proof of Lemma 4.7. We will prove this result by induction. For k = 1, the result is true by Lemma 2.4 in [4]. Let us assume that the result is true for k = n − 1. Fix a β0 in T. 1 Let β0 n be an n-th root of β0 and ω is a primitive n-th root of unity. Eq. (4.3) holds for n−1 1 each β = β0 n ω k for k = 0, 1, . . . , n − 1. Since k=0 ω jk = 0 for 1  j  n − 1, summing all these equations we have n−1 

 1  2 Re β0 n ω k X1 + β0 n ω 2k X2 + · · · + β0 Xn = Re β0 Xn  0.

k=0

Now varying β0 over T, from [4, Lemma 2.4], it follows that Xn = 0. By induction hypothesis, then we have X1 = X2 = · · · = Xn−1 = 0. 2 The following lemma will be useful in characterizing pure Γn -isometry. Lemma 4.8. Let Φ1 , . . . , Φn be functions in L∞ L(E) and MΦi , i = 1, . . . , n, denotes the corresponding multiplication operator on L2 (E). Then (MΦ1 , . . . , MΦn ) is a Γn -contraction if and only if (Φ1 (z), . . . , Φn (z)) is a Γn -contraction for all z in T. Proof. Note that for MΨ  = Ψ ∞ for Ψ ∈ L∞ L(E). By definition, (MΦ1 , . . . , MΦn ) is a Γn -contraction if and only if   p(MΦ , . . . , MΦ )  p∞,Γ , (4.6) 1 n n for all polynomials p ∈ C[z1 , . . . , zn ]. Since p(MΦ1 , . . . , MΦn ) = Mp(Φ1 ,...,Φn ) and MΨ  = Ψ ∞ := supz∈T Ψ (z) for Ψ ∈ L∞ L(E), we have (4.6) is true if and only if    p Φ1 (z), . . . , Φn (z)   p∞,Γ , n which completes the proof. 2

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Remark 4.9. An interesting case in the lemma above, is when dim E = 1. In this case, (Φ1 (z), . . . , Φn (z)) is a Γn -contraction means that (Φ1 (z), . . . , Φn (z)) is in Γn which is true as Γn is polynomially convex. Theorem 4.10. Let Si , i = 1, . . . , n, be commuting operators on a Hilbert space H. Then (S1 , . . . , Sn ) is a pure Γn -isometry if and only if there exist a separable Hilbert space E and a unitary operator U : H → H 2 (E) and functions Φ1 , . . . , Φn−1 in H ∞ L(E) and operators Ai ∈ L(E), i = 1, . . . , n − 1 such that (i) Si = U ∗ MΦi U , i = 1, . . . , n − 1, Sn = U ∗ Mz U ; (ii) (γ1 Φ1 (z), . . . , γn−1 Φn−1 (z)) is a Γn−1 -contraction for all z in T, where γi = n−i n for i = 1, . . . , n − 1; (iii) Φi (z) = Ai + A∗n−i z for 1  i  n − 1; (iv) [Ai , Aj ] = 0 and [Ai , A∗n−j ] = [Aj , A∗n−i ] for 1  i, j  n − 1, where [P, Q] = P Q − QP for two operators P , Q. Proof. Suppose (S1 , . . . , Sn ) is a pure Γn -isometry. By definition, there exist a Hilbert space K and a Γn -unitary (S˜1 , . . . , S˜n ) such that H ⊂ K is a common invariant subspace of S˜i ’s and Si = S˜i |H , i = 1, . . . , n. From Theorem 4.2, it follows that S˜n∗ S˜n = I and ∗ , 1  i, j  n − 1. By compression of S˜i to H, we have S˜n∗ S˜i = S˜n−i Sn∗ Sn = I

∗ and Sn∗ Si = Sn−i ,

1  i  n − 1.

Since Sn is a pure isometry and H is separable, there exists a unitary operator U : H → H 2 (E), for some separable Hilbert space E, such that Sn = U ∗ Mz U , where Mz is the shift operator on H 2 (E). Since Si ’s commute with Sn , there exist Φi ∈ H ∞ L(E) such that Si = U ∗ MΦi U for 1  i  n − 1. As (S1 , . . . , Sn ) is a Γn -contraction, from ∗ Lemma 2.7 and Lemma 4.8 part (ii) follows. The relations Sn∗ Si = Sn−i yield Mz¯MΦi = MΦ∗n−i , Let Φ(z) =

 n0

Cn z n , Ψ (z) = C0 z¯ + C1 +





n0

1  i  n − 1.

Dn z n be in H ∞ L(E). Then Mz¯MΦ = MΨ∗ implies

Cn z n−1 = D0∗ + D1∗ z¯ +

n2



Dn∗ z¯n

n2

for all z ∈ T and by comparing coefficients, we get C0 = D1∗

and C1 = D0∗ .

This gives that each Φi (z) is of the form Ai + Bi z for some Ai , Bi ∈ L(E), where ∗ for 1  i  n − 1, which is part (iii). Now since Si ’s commute, we have Ai = Bn−i MΦi MΦj = MΦj MΦi and consequently

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     Ai + A∗n−i z Aj + A∗n−j z = Aj + A∗n−j z Ai + A∗n−i z

for 1  i, j  n − 1 and for all z ∈ T. Comparing the constant term and the coefficients of z, we get part (iv). Conversely, suppose (S1 , . . . , Sn ) be the n-tuple satisfying conditions (i) to (iv). Consider the n-tuple (MΦ1 , . . . , MΦn−1 , Mz ) of multiplication operators on L2 (E) with symbols Φ1 , . . . , Φn−1 , z respectively. From condition (iii) and (iv), it follows that MΦi ’s commute with each other. Part of condition (iii) shows that Mz¯MΦi = MΦ∗n−i by repeating calculations similar to above. Thus, it is easy to see from part (ii) of Theorem 4.2, that (MΦ1 , . . . , MΦn−1 , Mz ) is a Γn -unitary and so is (U ∗ MΦ1 U, . . . , U ∗ MΦn−1 U, U ∗ Mz U ). Now, Si ’s, 1  i  n, are the restrictions to the common invariant subspace H 2 (E) of (MΦ1 , . . . , MΦn−1 , Mz ) and hence (S1 , . . . , Sn ) is a Γn -isometry. Since Sn is a shift, (S1 , . . . , Sn ) is a pure Γn -isometry. 2 Next, we obtain a characterization for Γn -isometry analogous to the Wold decomposition in terms of Γn -unitaries and pure Γn -isometries using Theorem 4.2 and Theorem 4.10. We prove this following Agler and Young [2]. Specifically, Lemma 2.5 from [2] will also be used in our proof. Lemma 4.11. (See [2, Lemma 2.5].) Let U and V be a unitary and a pure isometry on Hilbert spaces H1 , H2 respectively, and let T : H1 → H2 be an operator such that T U = V T . Then T = 0. Theorem 4.12. Let Si , i = 1, . . . , n, be commuting operators on a Hilbert space H. The following are equivalent: (i) (S1 , . . . , Sn ) is a Γn -isometry; (ii) There exists an orthogonal decomposition H = H1 ⊕ H2 into common reducing subspaces of Si ’s, i = 1, . . . , n such that (S1 |H1 , . . . , Sn |H1 ) is a Γn -unitary and (S1 |H2 , . . . , Sn |H2 ) is a pure Γn -isometry; ∗ (iii) Sn∗ Sn = I, Sn∗ Si = Sn−i and (γ1 S1 , . . . , γn−1 Sn−1 ) is a Γn−1 -contraction, where n−i γi = n for i = 1, . . . , n − 1. (iv) (γ1 S1 , . . . , γn−1 Sn−1 ) is a Γn−1 -contraction, where γi = n−i n for i = 1, . . . , n − 1 and for all w ∈ T, ρn (wS1 , w2 S2 , . . . , wn Sn ) = 0. Proof. Suppose (i) holds. By definition, there exists (S˜1 , . . . , S˜n ), a Γn -unitary on K containing H such that H is an invariant subspace of S˜i ’s and Si ’s are restrictions of S˜i ’s to H. From Theorem 4.2, it follows that S˜i ’s are satisfying the relations: S˜n∗ S˜n = I, S˜∗ S˜i = S˜∗ and (γ1 S˜1 , . . . , γn−1 S˜n−1 ) is a Γn−1 -contraction, where γi = n−i for n

n−i

n

i = 1, . . . , n − 1. Compressing to the common invariant subspace H and by part 1 of Remark 2.14, we obtain Sn∗ Sn = I,

∗ Sn∗ Si = Sn−i

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and (γ1 S1 , . . . , γn−1 Sn−1 ) is a Γn−1 -contraction, where γi = n−i n for i = 1, . . . , n − 1. Thus (i) implies (iii). Suppose (iii) holds. By Wold decomposition, we may write Sn = U ⊕V on H = H1 ⊕H2 where H1 , H2 are reducing subspaces for Sn , U is unitary and V is pure isometry. Let us write  Si =

S11

(i)

S12

(i)

S22

S21

(i)



(i)

(i)

with respect to this decomposition, where Sjk is a bounded operator from Hk to Hj . Since Sn Si = Si Sn , we have 

U S11

(i)

U S12

(i)

V S22

V S21

(i)



(i)

 =

S11 U

(i)

S12 V

(i)

S22 V

S21 U

(i) (i)

 i = 1, . . . , n − 1.

,

Thus, S21 U = V S21 and hence by Lemma 4.11, S21 = 0, i = 1, . . . , n − 1. Now Sn∗ Si = ∗ Sn−i gives (i)

(i)



(i)

U ∗ S11

U ∗ S12

0

V ∗ S22

(i)

(i) (i)



 =

(n−i)∗

S11

(n−i)∗

S12

0



(n−i)∗

S22

,

i = 1, . . . , n − 1.

(i)

It follows that S12 = 0, i = 1, . . . , n − 1. So H1 , H2 are common reducing subspaces (i) (n−i)∗ for S1 , . . . , Sn . From the matrix equation above, we have U ∗ S11 = S11 , i = 1, . . . , n − 1. Thus by part 1 of Remark 2.14 and part (ii) of Theorem 4.2, it follows that (1) (n−1) (S11 , . . . , S11 , U ) is a Γn -unitary. (1) (n−1) We now require to show that (S22 , . . . , S22 , V ) is a pure Γn -isometry. Since V is a pure isometry H is separable, we can identify it with the shift operator Mz on the (i) space of vector-valued functions H 2 (E), for some separable Hilbert space E. Since S22 ’s (i) ∞ commute with V , there exist Φi ∈ H L(E) such that S22 can be identified with MΦi for 1  i  n − 1. As (S1 , . . . , Sn ) is a Γn -contraction, from Lemma 2.7 and Lemma 4.8, (1) (n−1) it follows that (γ1 S22 , . . . , γn−1 S22 ) is a Γn−1 -contraction, where γi = n−i n for i = (n−i)∗ ∗ (i) 1, . . . , n − 1. The relations V S22 = S22 yield Mz¯MΦi = MΦ∗n−i ,

1  i  n − 1.

Calculations similar to the first part of the proof of Theorem 4.10, we get each Φi (z) is of (i) the form Ai + A∗n−i z for some Ai ’s in L(E), i = 1, . . . , n − 1. Now since S22 ’s commute, we have [Ai , Aj ] = 0 and

    Ai , A∗n−j = Aj , A∗n−i

for 1  i, j  n − 1.

Hence, by Theorem 4.10, (S1 |H2 , . . . , Sn |H2 ) is a pure Γn -isometry. Thus (iii) implies (ii).

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It is easy to see that (ii) implies (i). We will now prove (iii) is equivalent to (iv). As we have seen in Eq. (4.1), for w ∈ T, one could write n−k

2 ]  [   n−1

   2 n ∗ ρn wS1 , w S2 , . . . , w Sn = (−1)k n − (2i + k) wk Si∗ Si+k − Sn−k−i Sn−i

k=1 i=0 n−k

+

n−1 2 ]  [

 k ∗  ∗ (−1)k n − (2i + k) w ¯ Si+k Si − Sn−i Sn−k−i

k=1 i=0 n

+

[2] 

  ∗ (n − 2i) Si∗ Si − Sn−i Sn−i

i=0

where [x] denotes the largest integer less than or equal to x. Now if (iii) holds, as Si ’s commute, we have ∗ Si∗ Sj = Sn∗ Sn−i Sj = Sn∗ Sj Sn−i = Sn−j Sn−i ,

1  i, j  n − 1.

(4.7)

This shows ρn vanishes and hence (iii) implies (iv). Now if (iv) holds, multiplying by wk for each k = 0, 1, . . . , n − 1 and integrating over T, we have n

[2]    ∗ (n − 2i) Si∗ Si − Sn−i Sn−i = 0

(4.8)

i=0

and [ n−k 2 ]



n − (2i + k)



 ∗ ∗ Si+k Si − Sn−i Sn−k−i = 0

(4.9)

i=0

for each k = 1, . . . , n − 1. From Eq. (4.9), similar calculations as in Remark 4.6 will give ∗ Sn−i = Sn∗ Si for all i  n − 1. From Eq. (4.8) and calculations similar to (4.7), it follows that Sn∗ Sn = I. This completes the proof. 2 Corollary 4.13. Let Si , i = 1, . . . , n, be commuting operators on a Hilbert space H. Then (S1 , . . . , Sn ) is a Γn -co-isometry if and only if Sn Sn∗ = I,

Sn Si∗ = Sn−i

∗ and (γ1 S1∗ , . . . , γn−1 Sn−1 ) is a Γn−1 -contraction, where γi =

n−i n

for i = 1, . . . , n − 1.

5. Characterization of invariant subspaces for Γn -isometries This section is devoted to the characterization of the joint invariant subspaces of pure Γn -isometries. Similar characterization for pure Γ2 -isometries appears in [17].

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In the light of Theorem 4.10, we start with a characterization of pure Γn -isometries in terms of the parameters associated with them. For simplicity of notation, let (MΦ , Mz ) denote the n-tuple of multiplication operators (MΦ1 , . . . , MΦn−1 , Mz ) on H 2 (E), where Φi ∈ H ∞ L(E), i = 1, . . . , n − 1. Throughout this section, we will be using the canonical identification of H 2 (E) with H 2 ⊗ E by the map z n ξ → z n ⊗ ξ, where n ∈ N ∪ {0} and ξ ∈ E, whenever necessary. Theorem 5.1. Let Φi (z) = Ai + A∗n−i z and Φ˜i (z) = A˜i + A˜∗n−i z be in H ∞ L(E) for some Ai ∈ L(E) and A˜i ∈ L(F) respectively, i = 1, . . . , n − 1. Then the n-tuple (MΦ , Mz ) on H 2 (E) is unitarily equivalent to the n-tuple (MΦ˜ , Mz ) on H 2 (F) if and only if the (n − 1)-tuples (A1 , . . . , An−1 ) and (A˜1 , . . . , A˜n−1 ) are unitarily equivalent. Proof. Suppose the n-tuple (MΦ , Mz ) on H 2 (E) is unitarily equivalent to the n-tuple ˜i ) by IH 2 ⊗Ai +Mz ⊗A∗ . (MΦ˜ , Mz ) on H 2 (F). We can identify the map Φi (similarly Φ n−i So, there exists a unitary U : H 2 ⊗ E → H 2 ⊗ F such that   U IH 2 ⊗ Ai + Mz ⊗ A∗n−i U ∗ = IH 2 ⊗ A˜i + Mz ⊗ A˜∗n−i ,

i = 1, . . . , n − 1,

(5.1)

and U (Mz ⊗ IE )U ∗ = Mz ⊗ IF .

(5.2)

˜ : E → F such that U = IH 2 ⊗ U ˜. From Eq. (5.2), it follows that there exists a unitary U Consequently, Eq. (5.1) can be written as ˜ A∗ z U ˜ ∗ z = A˜i + A˜∗ z, ˜ Ai U ˜∗ + U U n−i n−i

i = 1, . . . , n − 1,

˜ Ai U ˜ ∗ = A˜i , i = 1, . . . , n − 1 for all z ∈ T. Hence comparing the coefficients, we obtain U which completes the proof in forward direction. ˜ : E → F that intertwines Ai and A˜i , that Conversely, suppose there exists a unitary U ∗ ˜ ˜ ˜ is, U Ai U = Ai for each i = 1, . . . , n − 1. Let U : H 2 ⊗ E → H 2 ⊗ F be the map defined ˜ . Clearly, U is a unitary and from the computations similar to above, it by U = IH 2 ⊗ U is easy to see that U intertwines MΦi with MΦ˜i for each i = 1, . . . , n − 1, and Mz . This completes the proof. 2 In the following corollary we express the theorem above in terms of pure Γn -isometries. Corollary 5.2. Let (S1 , . . . , Sn ) and (S˜1 , . . . , S˜n ) be a pair of pure Γn -isometries. Then (S1 , . . . , Sn ) and (S˜1 , . . . , S˜n ) are unitarily equivalent if and only if the (n − 1)-tuples ∗ ˜∗ ˜ ˜∗ n−1 − Si Sn∗ )n−1 (Sn−i i=1 and (Sn−i − Si Sn )i=1 are unitary equivalent. Proof. Let the pure Γn -isometry (S1 , . . . , Sn ) is equivalent to (MΦ , Mz ) where Φi (z) = Ai + A∗n−i z be in H ∞ L(E) for some Ai ∈ L(E), i = 1, . . . , n − 1 satisfying conditions

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∗ (ii) to (iv) in Theorem 4.2. Clearly the (n − 1)-tuple (Sn−i − Si Sn∗ )n−1 i=1 is equivalent to n−1 ∗ (MΦn−i − MΦi Mz¯)i=1 . Note that with the canonical identification we have

 ∗   MΦ∗n−i − MΦi Mz¯ = IH 2 ⊗ An−i + Mz ⊗ A∗i − IH 2 ⊗ Ai + Mz ⊗ A∗n−i (Mz¯ ⊗ IE ) = (IH 2 − Mz Mz¯) ⊗ A∗n−i = PC ⊗ A∗n−i where PC is the orthogonal projection from H 2 to the scalars in H 2 . Thus it follows for Theorem 5.1 that unitary equivalence of the n-tuple (MΦ , Mz ) is determined by unitary equivalence of the (n − 1)-tuple (MΦ∗n−i − MΦi Mz¯)n−1 i=1 , the unitary equivalence ∗ − Si Sn∗ )n−1 . This completes the proof. 2 of (S1 , . . . , Sn ) is determined by (Sn−i i=1 Characterization of invariant subspaces of Γn -isometries essentially boils down to that of pure Γn -isometries due to Wold type decomposition in Theorem 4.12 which is again the same as characterizing invariant subspace for the associated model space obtained in Theorem 4.10. The following theorem discusses this issue. A closed subspace M = {0} of H 2 (E) is said to be (MΦ , Mz )-invariant if M is invariant under MΦi and Mz for all i = 1, . . . , n − 1. Let M be a non-zero closed subspace of H 2 (E∗ ). It follows from the Beurling–Lax– Halmos theorem that M is invariant under Mz if and only if there exist a Hilbert space E and an inner function Θ ∈ H ∞ L(E, E∗ ) (Θ is an isometry almost everywhere on T) such that M = MΘ H 2 (E). Theorem 5.3. Let M be a non-zero closed subspace of H 2 (E∗ ). Then M is an invariant subspace of a pure isometry (MΦ , Mz ), Φi ∈ H ∞ L(E∗ ), i = 1, . . . , n − 1, if and only if there exist Ψi ∈ H ∞ L(E), i = 1, . . . , n − 1, such that (MΨ , Mz ) is a pure Γn -isometry on H 2 (E) and Φi Θ = ΘΨi ,

i = 1, . . . , n − 1,

where Θ ∈ H ∞ L(E, E∗ ) is the Beurling–Lax–Halmos representation of M. Proof. We will prove only the forward direction as the other part is easy to see by similar arguments. Let M is invariant under (MΦ , Mz ). Thus, in particular, M is invariant under Mz and hence the Beurling–Lax–Halmos representation of M is ΘH 2 (E) where Θ ∈ H ∞ L(E, E∗ ) is an inner multiplier. We also have MΦi ΘH 2 (E) ⊆ ΘH 2 (E) for each i = 1, . . . , n − 1. Thus, there exist Ψi ’s, i = 1, . . . , n − 1 in H ∞ L(E) such that MΦi MΘ = MΘ MΨi , that is, Φi Θ = ΘΨi , i = 1, . . . , n−1. Since Φi ’s commute, we have MΘ MΨi MΨj = MΦi MΦj MΘ = MΦj MΦi MΘ = MΘ MΨj MΨi and hence Ψi Ψj = Ψj Ψi , i, j = 1, . . . , n − 1. Furthermore, we have

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p(MΦ1 , . . . , MΦn−1 )MΘ = MΘ p(MΨ1 , . . . , MΨn−1 ) for all polynomials p ∈ C[z1 , . . . , zn−1 ]. Therefore,     p(γ1 MΨ1 , . . . , γn−1 MΨn−1 )  MΘ∗ p(γ1 MΦ1 , . . . , γn−1 MΦn−1 )MΘ   p∞,Γn−1 , for all polynomials p ∈ C[z1 , . . . , zn−1 ] and γi = rem 4.10, we also note that

n−i n

for i = 1, . . . , n − 1. Using Theo-

Mz¯MΨi = Mz¯MΘ∗ MΦi MΘ = MΘ∗ Mz¯MΦi MΘ = MΘ∗ MΦ∗n−i MΘ = MΨ∗n−i , for i = 1, . . . , n − 1. From the observations made above and by Theorem 4.10, it follows that (MΨ , Mz ) is a pure Γn -isometry on H 2 (E) and this completes the proof. 2 Acknowledgments We thank the anonymous reviewers for valuable suggestions. References [1] J. Agler, N.J. Young, Operators having the symmetrized bidisc as a spectral set, Proc. Edinb. Math. Soc. (2) 43 (1) (2000) 195–210. [2] J. Agler, N.J. Young, A model theory for Γ -contractions, J. Operator Theory 49 (2003) 45–60. [3] J. Agler, N.J. Young, The magic functions and automorphisms of a domain, Complex Anal. Oper. Theory 2 (3) (2008) 383–404. [4] T. Bhattacharyya, S. Pal, S. Shyam Roy, Dilations of Γ -contractions by solving operator equations, Adv. Math. 230 (2) (2012) 577–606. [5] R. Curto, N. Salinas, Generalized Bergman kernels and the Cowen–Douglas theory, Amer. J. Math. 106 (2) (1984) 447–488. [6] A. Edigarian, W. Zwonek, Geometry of the symmetrized polydisc, Arch. Math. 84 (2005) 364–374. [7] J.P. Escofier, Galois Theory, Grad. Texts in Math., vol. 204, Springer-Verlag, New York, 2001, translated from the 1997 French original by Leila Schneps. [8] A. Grinshpan, D.S. Kaliuzhnyi-Verbovetskyi, V. Vinnikov, H. Woerdeman, Classes of tuples of commuting contractions satisfying the multivariable von Neumann inequality, J. Funct. Anal. 256 (9) (2009) 3035–3054. [9] P. Henrici, Applied and Computational Complex Analysis, vol. 1. Power Series Integration Conformal Mapping Location of Zeros, reprint of the 1974 original, Wiley Classics Library, A Wiley– Interscience Publication, John Wiley and Sons, Inc., New York, 1988. [10] G. James, M. Liebeck, Representations and Characters of Groups, second edition, Cambridge University Press, New York, 2001. [11] D.E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups, reprint of the second (1950) edition, AMS Chelsea Publishing, Providence, RI, 2006. [12] M. Marden, Geometry of Polynomials, second edition, Math. Surveys Monogr., vol. 3, American Mathematical Society, Providence, RI, 1966. [13] G. Misra, S. Shyam Roy, G. Zhang, Reproducing kernel for a class of weighted Bergman spaces on the symmetrized polydisc, Proc. Amer. Math. Soc. 141 (7) (2013) 2361–2370. [14] D.J. Ogle, Operator and function theory of the symmetrized polydisc, PhD thesis, http://www1. maths.leeds.ac.uk/~nicholas/abstracts/oglethesis.pdf. [15] V. Paulsen, Completely Bounded Maps and Operator Algebras, Cambridge Stud. Adv. Math., vol. 78, Cambridge University Press, Cambridge, 2002.

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