Toeplitz Operators on the Symmetrized Bidisc

Bhattacharyya, Tirthankar; Das, B. Krishna; Sau, Haripada

doi:10.1093/imrn/rnz333

Cited by 11 publications

(14 citation statements)

References 25 publications

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“…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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confidence: 81%

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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confidence: 81%

Algebraic Properties of Toeplitz operators on the symmetrized polydisk

Das¹,

Sau²

2021

Preprint

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“…This kernel is known to have all the properties to be referred to as the Szegö kernel for G -see the papers [36,21,18]. One of the criteria for the solvability of a Pick problem in G is the following.…”

Section: Extremal Problems and The Uniqueness Varietymentioning

confidence: 99%

Distinguished varieties and the Nevanlinna-Pick interpolation problem on the symmetrized bidisk

Das,

Kumar,

2021

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We study the uniqueness of the solutions of a solvable Pick interpolation problem in the symmetrized bidiskThe uniqueness set is the largest set in G where all the solutions to a solvable Pick problem coincide. For a solvable Pick problem in G, there is a canonical construction of an algebraic variety, which coincides with the uniqueness set in G. The algebraic variety is called the uniqueness variety. In the first main result of this paper, we show that if an N -point solvable Pick problem is such that it has no solutions of supremum norm (over G) less than one and that each of the (N − 1)-point subproblems has a solution of supremum norm less than one, then the uniqueness variety corresponding to the N -point problem contains a distinguished variety containing all the initial nodes. Here, a distinguished variety is an algebraic variety that intersects the domain G and exits through its distinguished boundary. The proof of the first main result requires a thorough understanding of distinguished varieties. Indeed, this article is as much a study of the uniqueness varieties as it is about distinguished varieties. We obtain a complete algebraic and geometric characterizations of distinguished varieties.

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“…We shall require some results from this theory, many of which can be found in [12] and [6, Appendix A]. These domains have a rich complex geometry and function theory, as well as applications to operator theory: see, besides many other papers, [4,[7][8][9]12,[14][15][16]].…”

Section: Theorem 14 If D Is a Purely Balanced Geodesic In G Then (Dmentioning

confidence: 99%

“…Indeed, R − is the polynomially convex hull of E 9. Equivalently, a geodesic D is flat if and only if D − ∩ E = ∅.…”

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Intrinsic Directions, Orthogonality, and Distinguished Geodesics in the Symmetrized Bidisc

2021

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The symmetrized bidisc $$\begin{aligned} G {\mathop {=}\limits ^\mathrm{{def}}}\{(z+w,zw):|z|<1,\quad |w|<1\}, \end{aligned}$$ G = def { ( z + w , z w ) : | z | < 1 , | w | < 1 } , under the Carathéodory metric, is a complex Finsler space of cohomogeneity 1 in which the geodesics, both real and complex, enjoy a rich geometry. As a Finsler manifold, G does not admit a natural notion of angle, but we nevertheless show that there is a notion of orthogonality. The complex tangent bundle TG splits naturally into the direct sum of two line bundles, which we call the sharp and flat bundles, and which are geometrically defined and therefore covariant under automorphisms of G. Through every point of G, there is a unique complex geodesic of G in the flat direction, having the form $$\begin{aligned} F^\beta {\mathop {=}\limits ^\mathrm{{def}}}\{(\beta +{\bar{\beta }} z,z)\ : z\in \mathbb {D}\} \end{aligned}$$ F β = def { ( β + β ¯ z , z ) : z ∈ D } for some $$\beta \in \mathbb {D}$$ β ∈ D , and called a flat geodesic. We say that a complex geodesic Dis orthogonal to a flat geodesic F if D meets F at a point $$\lambda $$ λ and the complex tangent space $$T_\lambda D$$ T λ D at $$\lambda $$ λ is in the sharp direction at $$\lambda $$ λ . We prove that a geodesic D has the closest point property with respect to a flat geodesic F if and only if D is orthogonal to F in the above sense. Moreover, G is foliated by the geodesics in G that are orthogonal to a fixed flat geodesic F.

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Toeplitz Operators on the Symmetrized Bidisc

Cited by 11 publications

References 25 publications

Algebraic Properties of Toeplitz operators on the symmetrized polydisk

Algebraic Properties of Toeplitz operators on the symmetrized polydisk

Distinguished varieties and the Nevanlinna-Pick interpolation problem on the symmetrized bidisk

Intrinsic Directions, Orthogonality, and Distinguished Geodesics in the Symmetrized Bidisc

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