Realization of functions on the symmetrized bidisc

Agler, Jim; Young, N. J.

doi:10.1016/j.jmaa.2017.04.003

Cited by 32 publications

(17 citation statements)

References 15 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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“…Recall the following model formula [6,Definition 2.1]. In this section we use the symbols λ, µ for points of G and co-ordinates λ = (s, p).…”

Section: The Model Formula For the Symmetrized Bidiscmentioning

confidence: 99%

“…For the proof see [6,Theorem 2.2]. From a G-model of a function ϕ ∈ S(G) one may easily proceed by means of a standard lurking isometry argument to a realization formula of the form…”

Section: The Model Formula For the Symmetrized Bidiscmentioning

confidence: 99%

“…The main tool we use is a model formula for analytic functions from G to the closed unit disc D − proved in [5] and stated below as Definition 2.2 and Theorem 2.3. Model formulae and realization formulae for a class of functions are essentially equivalent: one can pass back and forth between them by standard methods (algebraic manipulation in one direction, lurking isometry arguments in the other).…”

Section: Introductionmentioning

confidence: 99%

“…The main tool we use in Section 3 is a model formula for analytic functions from G to the closed unit disc D proved in [6] and stated below as Definition 1.1 and Theorem 1.2. We also use some basic properties of linear fractional transformations with operator coefficients.…”

Section: Introductionmentioning

confidence: 99%

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