Toeplitz Operators Research Articles

On Compactness of Products of Toeplitz Operators

AbstractWe study compactness of product of Toeplitz operators with symbols continuous on the closure of the polydisc in terms of behavior of the symbols on the boundary. For certain classes of symbols f and g, we show that $$T_fT_g$$ T f T g is compact if and only if fg vanishes on the boundary. We provide examples to show that for more general symbols, the vanishing of fg on the whole polydisc might not imply the compactness of $$T_fT_g$$ T f T g . On the other hand, the reverse direction is closely related to the zero product problem for Toeplitz operators on the unit disc, which is still open.

A note on the space of all Toeplitz operators

Abstract We study Toeplitz operators on the space of all real analytic functions on the real line and the space of all holomorphic functions on finitely connected domains in the complex plane. In both cases, we show that the space of all Toeplitz operators is isomorphic, when equipped with the topology of uniform convergence on bounded sets, with the symbol algebra. This is surprising in view of our previous results, since we showed that the symbol map is not continuous in this topology on the algebra generated by all Toeplitz operators. We also show that in the case of the Fréchet space of all holomorphic functions on a finitely connected domain in the complex plane, the commutator ideal is dense in the algebra generated by all Toeplitz operators in the topology of uniform convergence on bounded sets.

On the essential norms of Toeplitz operators on abstract Hardy spaces built upon Banach function spaces

Let X be a Banach function space over the unit circle such that the Riesz projection P is bounded on X and let H[X] be the abstract Hardy space built upon X. We show that the essential norm of the Toeplitz operator T(a):H[X]→H[X]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T(a):H[X]\\rightarrow H[X]$$\\end{document} coincides with ‖a‖L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Vert a\\Vert _{L^\\infty }$$\\end{document} for every a∈C+H∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$a\\in C+H^\\infty $$\\end{document} if and only if the essential norm of the backward shift operator T(e-1):H[X]→H[X]\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$T(\ extbf{e}_{-1}):H[X]\\rightarrow H[X]$$\\end{document} is equal to one, where e-1(z)=z-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extbf{e}_{-1}(z)=z^{-1}$$\\end{document}. This result extends an observation by Böttcher, Krupnik, and Silbermann for the case of classical Hardy spaces.

Examining Hyponormality in Toeplitz Operators: Non-Harmonic Symbols on Weighted Bergman Spaces

Examining Hyponormality in Toeplitz Operators: Non-Harmonic Symbols on Weighted Bergman Spaces

Quantum Harmonic Analysis for Polyanalytic Fock Spaces

We develop the quantum harmonic analysis framework in the reducible setting and apply our findings to polyanalytic Fock spaces. In particular, we explain some phenomena observed in recent work by the second author and answer a few related open questions. For instance, we show that there exists a symbol such that the corresponding Toeplitz operator is unitary on the analytic Fock space but vanishes completely on one of the true polyanalytic Fock spaces. This follows directly from an explicit characterization of the kernel of the Toeplitz quantization, which we derive using quantum harmonic analysis. Moreover, we show that the Berezin transform is injective on the set of of Toeplitz operators. Finally, we provide several characterizations of the C1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {C}_1$$\\end{document}-algebra in terms of integral kernel estimates and essential commutants.

Numerical ranges of some 2-by-2 block Toeplitz operators with affine symbols via envelopes

Numerical ranges of some 2-by-2 block Toeplitz operators with affine symbols via envelopes

Toeplitz operators with symmetric, alternating and anti-symmetric separately radial symbols on the unit ball

Toeplitz operators with symmetric, alternating and anti-symmetric separately radial symbols on the unit ball

Wiener-Hopf Indices of Unitary-Valued Functions on the Imaginary Axis

This paper is concerned with the Wiener-Hopf indices of unitary-valued rational matrix functions on the imaginary axis. These indices play a role in the Fredholm theory for Wiener-Hopf integral operators. Our main result gives formulas for the Wiener-Hopf indices in terms of the matrices appearing in realizations of the factors in a Douglas-Shapiro-Shields factorization of the unitary-valued function. Two approaches to this problem are presented: one direct approach using operator theoretic methods, and a second approach using the Cayley transform which allows to use results for an analogous problem regarding unitary-valued functions on the unit circle and corresponding Toeplitz operators.

C-symmetric Toeplitz operators on Hardy spaces

We characterize all the Toeplitz operators that are complex symmetric with respect to a class of conjugations induced by a permutation. Our results provide an affirmative answer to a conjecture from a paper of Chattopadhyay et al. (2023) [1].

Brown-Halmos type theorems for Toeplitz operators on the Bergman space of the upper half-plane

We investigate Brown-Halmos type theorems which consist in characterizing when the product of two Bergman space Toeplitz operators is a Toeplitz operator whose symbol is the product of their symbols. The main involved technique is the characterization of functions that are in the range of the Berezin transform, and those which have finite rank Berezin symbols. We establish the upper half-plane analogues of the results of P. Ahern and Ž. Čučković (2001), P. Ahern (2004), and N.V. Rao (2018). In particular, we consider two classes of Toeplitz operators, namely those with bounded harmonic symbols and those with bi-analytic or tri-analytic symbols. Due to the unbounded character of the domain, many challenging difficulties occur, requiring more careful analysis.

A remark on Toeplitz and Laurent operators acting on ℓ spaces with power weights

A remark on Toeplitz and Laurent operators acting on ℓ spaces with power weights

Asymptotics of block Toeplitz determinants with piecewise continuous symbols

AbstractWe determine the asymptotics of the block Toeplitz determinants as for matrix‐valued piecewise continuous functions with a finitely many jumps under mild additional conditions. In particular, we prove that where , , and are constants that depend on the matrix symbol and are described in our main results. Our approach is based on a new localization theorem for Toeplitz determinants, a new method of computing the Fredholm index of Toeplitz operators with piecewise continuous matrix‐valued symbols, and other operator theoretic methods. As an application of our results, we consider piecewise continuous symbols that arise in the study of entanglement entropy in quantum spin chain models.

Berezin transform of products of Toeplitz operators on the Hardy space

Let H 2 ( S ) H^2(S) be the Hardy space on the unit sphere in C n \mathbf {C}^n . We show that there are Toeplitz operators T f T_f and T g T_g on H 2 ( S ) H^2(S) such that the product T f T g T_fT_g is not compact and yet ‖ T f T g k z ‖ \|T_fT_gk_z\| tends to 0 0 as | z | → 1 |z| \rightarrow 1 . Consequently, the Berezin transform ⟨ T f T g k z , k z ⟩ \langle T_fT_gk_z,k_z\rangle tends to 0 0 as | z | → 1 |z| \rightarrow 1 .

Toeplitz operators between weighted Bergman spaces over the half-plane

Toeplitz operators between weighted Bergman spaces over the half-plane

On hyponormality and a commuting property of Toeplitz operators

On hyponormality and a commuting property of Toeplitz operators

Toeplitz operators on monomial polyhedra

Toeplitz operators on monomial polyhedra

Semi-commutants of Toeplitz Operators on Fock–Sobolev Space of Nonnegative Orders

Semi-commutants of Toeplitz Operators on Fock–Sobolev Space of Nonnegative Orders

Inverses of Toeplitz plus Hankel operators with generating matrix functions

Inverses of Toeplitz plus Hankel operators with generating matrix functions

Functional calculus and quantization commutes with reduction for Toeplitz operators on CR manifolds

Functional calculus and quantization commutes with reduction for Toeplitz operators on CR manifolds

Toeplitz operators and Hankel operators on a Bergman space with an exponential weight on the unit ball

Toeplitz operators and Hankel operators on a Bergman space with an exponential weight on the unit ball