$\Gamma$-Unitaries, Dilation and a Natural Example

Bhattacharyya, Tirthankar; Sau, Haripada

doi:10.4171/prims/53-2-2

Cited by 3 publications

(8 citation statements)

References 8 publications

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“…The operator Y that appeared in Example 16 and also in Theorem 29 while characterizing compact operators on H 2 (G) is the adjoint of the fundamental operator of the Γ-coisometry (T * s , T * p ). This follows from [11] and was discussed in detail in Section 5 of [11].…”

Section: Commutant Lifting and Dual Toeplitz Operatorsmentioning

confidence: 99%

“…The content of this lemma is from [11]. We have included it here for the sake of completeness as well and for a more succinct presentation than in [11]. By virtue of the isomorphisms U and U 1 described above, we have the following commutative diagram:…”

Section: Hardy Space and Boundary Valuesmentioning

confidence: 99%

Section: Is a Toeplitz Operator;mentioning

confidence: 99%

“…The content of this lemma is from [11]. We have included it here for the sake of completeness as well and for a more succinct presentation than in [11].…”

Section: Remarkmentioning

confidence: 99%

“…It has been observed over the last decade that the symmetrized bidisc enjoys some one dimensional phenomena, e.g., the automorphism group of G is isomorphic to the automorphism group of the unit disc (Theorem 4.1 of [4]), the minimal normal boundary dilation of this domain acts on the minimal normal boundary dilation of the unit disc [11]. However, the following example shows that the Coburn Alternative, which has several useful consequences in the study of Toeplitz operators on the unit disc, fails to hold true in the symmetrized bidisc.…”

Section: Theorem 25 the Only Compact Toeplitz Operator On The Symmetr...mentioning

confidence: 99%

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Toeplitz Operators on the Symmetrized Bidisc

Bhattacharyya

Das

2020

International Mathematics Research Notices

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The symmetrized bidisc has grabbed a great deal of attention of late because of its rich structure both in the context of function theory and in the context of operator theory. Toeplitz operators on this domain have not been discussed so far. The distinguished boundary bΓ of the symmetrized bidisc is topologically identifiable with the Mobius strip and it is natural to consider bounded measurable functions there. In this article, we show that there is a natural Hilbert space H 2 (G). We describe three isomorphic copies of this space. The L ∞ functions on bΓ induce Toeplitz operators on this space. Such Toeplitz operators can be characterized through a couple of relations that they have to satisfy with respect to the co-ordinate multiplications on the space H 2 (G) which we call the Brown-Halmos relations. A number of results are obtained about the Toeplitz operators which bring out the similarities and the differences with the theory of Toeplitz operators on the disc as well as the bidisc. We show that the Coburn alternative fails, for example. However, the compact perturbations of Toeplitz operators are precisely the asymptotic Toeplitz operators. This requires us to find a characterization of compact operators on the Hardy space H 2 (G). The only compact Toeplitz operator turns out to be the zero operator.Although operator theory on the symmetrized bidisc has now been studied for quite some time, often there are occasions when one has to develop a result that one needs. Such is the case we encountered in the study of dual Toeplitz operators in the last section of this paper. In that section, we produce a new result about a family of commuting Γisometries. Just like a Toeplitz operator is characterized by the Brown-Halmos relations with respect to the co-ordinate multiplications, an arbitrary bounded operator X which satisfies the Brown-Halmos relations with respect to a commuting family of Γ-isometries is a compression of a norm preserving Y acting on the space of minimal Γ-unitary extension of the family of isometries. Moreover, if X commutes with the Γ-isometries, then Y is an extension and commutes with the minimal Γ-unitary extensions. Thus, it is a commutant lifting theorem. This result is then applied to characterize a dual Toeplitz operator.2010 Mathematics Subject Classification. 47A13, 47A20, 47B35, 47B38, 46E20, 30H10.

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Section: Commutant Lifting and Dual Toeplitz Operatorsmentioning

confidence: 99%

Section: Hardy Space and Boundary Valuesmentioning

confidence: 99%

Section: Is a Toeplitz Operator;mentioning

confidence: 99%

“…The content of this lemma is from [11]. We have included it here for the sake of completeness as well and for a more succinct presentation than in [11].…”

Section: Remarkmentioning

confidence: 99%

Section: Theorem 25 the Only Compact Toeplitz Operator On The Symmetr...mentioning

confidence: 99%

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Toeplitz Operators on the Symmetrized Bidisc

Bhattacharyya

Das

2020

International Mathematics Research Notices

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Functional Models for Commuting Hilbert-Space Contractions

Ball

2020

Operator Theory, Operator Algebras and Their Interactions With Geometry and Topology

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We develop a Sz.-Nagy-Foias-type functional model for a commutative contractive operator tuple T = (T1, . . . , T d ) having T = T1 • • • T d equal to a completely nonunitary contraction. We identify additional invariants G ♯ , W ♯ in addition to the Sz.-Nagy-Foias characteristic function ΘT for the product operator T so that the combined triple (G ♯ , W ♯ , ΘT ) becomes a complete unitary invariant for the original operator tuple T . For the case d ≥ 3 in general there is no commutative isometric lift of T ; however there is a (not necessarily commutative) isometric lift having some additional structure so that, when compressed to the minimal isometric-lift space for the product operator T , generates a special kind of lift of T , herein called a pseudo-commutative contractive lift of T , which in turn leads to the functional model for T . This work has many parallels with recently developed model theories for symmetrized-bidisk contractions (commutative operator pairs (S, P ) having the symmetrized bidisk Γ as a spectral set) and for tetrablock contractions (commutative operator triples (A, B, P ) having the tetrablock domain E as a spectral set).

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Andô dilations for a pair of commuting contractions: two explicit constructions and functional models

2017

Preprint

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One of the most important results in operator theory is Andô's [3] generalization of dilation theory for a single contraction to a pair of commuting contractions acting on a Hilbert space. While there are two explicit constructions (Schäffer [29] and Douglas [18]) of the minimal isometric dilation of a single contraction, there was no such explicit construction of an Andô dilation for a commuting pair (T 1 , T 2 ) of contractions, except in some special cases [2,16,17]. In this paper, we give two new proofs of Andô's dilation theorem by giving both Schäffer-type and Douglas-type explicit constructions of an Andô dilation with function-theoretic interpretation, for the general case. The results, in particular, give a complete description of all possible factorizations of a given contraction T into the product of two commuting contractions. Unlike the one-variable case, two minimal Andô dilations need not be unitarily equivalent. However, we show that the compressions of the two Andô dilations constructed in this paper to the minimal dilation spaces of the contraction T 1 T 2 , are unitarily equivalent.In the special case when the product T = T 1 T 2 is pure, i.e., if T * n → 0 strongly, an Andô dilation was constructed recently in [17], which, as this paper will show, is a corollary to the Douglas-type construction. We also show that their construction in this special case can be derived from a previous result obtained in [28].We define a notion of characteristic triple for a pair of commuting contractions and a notion of coincidence for such triples. We prove that two pairs of commuting contractions with their products being pure contractions are unitarily equivalent if and only if their characteristic triples coincide. We also characterize triples which qualify as the characteristic triple for some pair (T 1 , T 2 ) of commuting contractions such that T 1 T 2 is a pure contraction.

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$\Gamma$-Unitaries, Dilation and a Natural Example

Cited by 3 publications

References 8 publications

Toeplitz Operators on the Symmetrized Bidisc

Toeplitz Operators on the Symmetrized Bidisc

Functional Models for Commuting Hilbert-Space Contractions

Andô dilations for a pair of commuting contractions: two explicit constructions and functional models

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