Holomorphic functions on the symmetrized bidisk – Realization, interpolation and extension

Bhattacharyya, Tirthankar; Sau, Haripada

doi:10.1016/j.jfa.2017.09.013

Cited by 22 publications

(33 citation statements)

References 28 publications

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“…The novelty of this domain arises from the fact that it exhibits one-dimensional behaviour at times (e.g., the automorphism group is the same as that of the unit disc in the plane) and behaves significantly different at times (e.g., a realization formula for a function in the unit ball of H ∞ (G) requires uncountably infinitely many "test functions", see [5] and [10]). The Toeplitz operators on this domain will highlight a few similarities and a lot of differences with the classical situation of Brown and Halmos as well as with later endeavours of the bidisc.…”

Section: Hardy Space and Boundary Valuesmentioning

confidence: 99%

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: Hardy Space and Boundary Valuesmentioning

confidence: 99%

Toeplitz Operators on the Symmetrized Bidisc

Bhattacharyya

Das

2020

International Mathematics Research Notices

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“…The same theorem is stated in [19]. In the event that A = H ∞ (V ) for a subset V of G, we can describe all (G, A)von Neumann sets.…”

Section: Chapter 14mentioning

confidence: 84%

Geodesics, Retracts, and the Norm-Preserving Extension Property in the Symmetrized Bidisc

2019

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Preface vii Chapter 1. Introduction Chapter 2. An overview Chapter 3. Extremal problems in the symmetrized bidisc G 3.1. The Carathéodory and Kobayashi extremal problems 3.2. The Carathéodory extremal problem Car(δ) for G 3.3. Five types of datum δ in G 3.4. The Kobayashi extremal problem Kob(δ) for G Chapter 4. Complex geodesics in G 4.1. Complex geodesics and datums in G 4.2. Uniqueness of complex geodesics for each datum in G 4.3. Flat C-geodesics 4.4. Rational Γ-inner functions Chapter 5. The retracts of G and the bidisc D 2 5.1. Retracts and geodesics of G 5.2. Retracts of D 2 5.3. Geodesics in G are varieties Chapter 6. Purely unbalanced and exceptional datums in G Chapter 7. A geometric classification of geodesics in G Chapter 8. Balanced geodesics in G Chapter 9. Geodesics and sets V with the norm-preserving extension property in G 9.1. V and Car(δ) 9.2. V and balanced datums 9.3. V and flat or royal datums Chapter 10. Anomalous sets R ∪ D with the norm-preserving extension property in G 10.1. Definitions and lemmas 10.2. The proof of the norm-preserving extension property for R ∪ D Chapter 11. V and a circular region R in the plane Chapter 12. Proof of the main theorem iii iv CONTENTS 12.1. Preliminary lemmas 12.2. σ : R → V is analytic on R − 12.3. Degree considerations Chapter 13. Sets in D 2 with the symmetric extension property Chapter 14. Applications to the theory of spectral sets Chapter 15. Anomalous sets with the norm-preserving extension property in some other domains Appendix A. Some useful facts about the symmetrized bidisc A.1. Basic properties of G and Γ A.2. Complex C-geodesics in G A.3. Automorphisms of G A.4. A trichotomy theorem A.5. Datums for which all Φ ω are extremal Appendix B. Types of geodesic: a crib and some cartoons B.1. Crib sheet B.2. Cartoons Bibliography Index

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“…Another criterion for the solvability of a finite interpolation problem in S (G) is given in [12,Theorem 6.1]. It is shown that, in the situation of Theorem 5.1, a desired interpolating function ϕ ∈ S (G) exists if and only if there exists a C(D − ) * -valued positive semidefinite kernel on {s 1 , .…”

Section: A Pick Theorem For Gmentioning

confidence: 99%

“…A generalization of the realization theory of the polydisc to much more general domains, based on test functions, has been developed by Dritschel, McCullough and others [13,14,10]. We thank a referee for the observation that a realization formula for functions in the Schur-Agler class of G can be derived from the 'abstract realization theorem' [13, Theorem 2.2] by the choice of the functions s → 2λs 2 − s 1 2 − λs 1 (for |λ| < 1) as the test functions on G. This procedure is essentially carried out in [12], where a realization formula somewhat similar to ours is given [12, Realization theorem, page 5]. However, this approach only yields a realization formula for the Schur-Agler class, not the Schur class, and so to prove Theorem 1.1 in this way one must invoke [4], implicitly utilizing the symmetrization argument we use in this paper.…”

Section: Introductionmentioning

confidence: 99%