Dual Toeplitz Operators on the Sphere Via Spherical Isometries

Didas, Michael; Eschmeier, Jörg

doi:10.1007/s00020-015-2232-7

Cited by 4 publications

(3 citation statements)

References 8 publications

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“…Dual Toeplitz operators have been studied for a while on the Bergman space of the unit disc D in [27] and on the Hardy space of the Euclidean ball B d in [17]. In our setting, consider the space and from the 2×2 matrix representations of the operators in concern.…”

Section: Commutant Lifting and Dual Toeplitz Operatorsmentioning

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: Commutant Lifting and Dual Toeplitz Operatorsmentioning

confidence: 99%

Toeplitz Operators on the Symmetrized Bidisc

Bhattacharyya

Das

2020

International Mathematics Research Notices

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“…Dual Toeplitz operators have been studied for a while on the Bergman space of the unit disc D in [30] and on the Hardy space of the Euclidean ball B d in [15]. In our setting, consider the space…”

Section: Dual Toeplitz Operatorsmentioning

confidence: 99%

Algebraic Properties of Toeplitz operators on the symmetrized polydisk

Das¹,

Sau²

2021

Preprint

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This paper is an effort to continue the legacy of the classically successful theory of Toeplitz operators on the Hardy space over the unit disk to a new domain in C d -the symmetrized polydisk. We obtain algebraic characterizations of Toeplitz operators, analytic Toeplitz operators, compact perturbation of Toeplitz operators and dual Toeplitz operators. We then revisit the operator theory of this domain considered first in [10], to study the generalized Toeplitz operators and find a commutant lifting type result. This is a continuation of a previous work done in [8].

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“…Dual Toeplitz operators. Dual Toeplitz operators have been studied for a while on the Bergman space of the unit disc D in [23] and on the Hardy space of the Euclidean ball B d in [10]. In our setting, consider the space By Theorem 14 and the fact that any operator that commutes with each of M s 1 , .…”

Section: 2mentioning

confidence: 99%

On certain Toeplitz operators and associated completely positive maps

Bhattacharyya¹,

Das²,

Sau³

2017

Preprint

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We study Toeplitz operators with respect to a commuting n-tuple of bounded operators which satisfies some additional conditions coming from complex geometry. Then we consider a particular such tuple on a function space. The algebra of Toeplitz operators with respect to that particular tuple becomes naturally homeomorphic to L ∞ of a certain compact subset of C n . Dual Toeplitz operators are characterized. En route, we prove an extension type theorem which is not only important for studying Toeplitz operators, but also has an independent interest because dilation theorems do not hold in general for n > 2.

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Dual Toeplitz Operators on the Sphere Via Spherical Isometries

Cited by 4 publications

References 8 publications

Toeplitz Operators on the Symmetrized Bidisc

Toeplitz Operators on the Symmetrized Bidisc

Algebraic Properties of Toeplitz operators on the symmetrized polydisk

On certain Toeplitz operators and associated completely positive maps

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