On the existence of eigenvalues of Toeplitz operators on planar regions

A study is made of the eigenvalues of self-adjoint Toeplitz operators on multiply connected planar regions having holes. The presence of eigenvalues is detected through an analysis of the zeros of translations of theta functions restricted to in .

     

Separately radial and radial Toeplitz operators on the projective space and representation theory

We consider separately radial (with corresponding group T ⁿ ) and radial (with corresponding group U(n)) symbols on the projective space P ⁿ (C), as well as the associated Toeplitz operators on the weighted Bergman spaces. It is known that the C*-algebras generated by each family of such Toeplitz operators are commutative (see R. Quiroga-Barranco and A. Sanchez-Nungaray (2011)). We present a new representation theoretic proof of such commutativity. Our method is easier and more enlightening as it shows that the commutativity of the C*-algebras is a consequence of the existence of multiplicity-free representations. Furthermore, our method shows how to extend the current formulas for the spectra of the corresponding Toeplitz operators to any closed group lying between T ⁿ and U(n).     

Bicommutants of Toeplitz operators

In this paper we discuss an unusual phenomenon in the context of Toeplitz operators in the Bergman space on the unit disc: If two Toeplitz operators commute with a quasihomogeneous Toeplitz operator, then they commute with each other. In the Bourbaki terminology, this result can be stated as follows: The commutant of a quasihomogeneous Toeplitz operator is equal to its bicommutant. Received: 11 March 2008     

Products of Toeplitz operators

An aspect of the theory of Toeplitz operators on generalised Hardy spaces is considered, namely, a necessary and sufficient condition on the symbols to ensure that the product of two Toeplitz operators is itself a Toeplitz operator. The answer to this question draws on many deep results of the theory of generalised Hardy spaces.     

Products of Toeplitz operators

A sufficient condition is found for the product of two Toeplitz operators to be a compact perturbation of a Toeplitz operator. The condition, which comprehends all previously known sufficient conditions, is shown to be necessary under additional hypotheses. The question whether the condition is necessary in general is left open.     

Asymptotics of the eigenvalues of Toeplitz forms

In this paper we obtain a refinement of Szegö's well-known theorem on the asymptotics of the eigenvalues of Toeplitz forms.Translated from Matematicheskie Zametki, Vol. 11, No. 2, pp. 151–158, February, 1972.     

New regions including eigenvalues of Toeplitz matrices

Two new eigenvalue inclusion regions for matrices with a constant main diagonal are given. We then apply these results to Toeplitz matrices, and obtain two regions including all eigenvalues of Toeplitz matrices. Furthermore, it is proved that the new regions are tighter than those in [Melman A. Ovals of Cassini for Toeplitz matrices, Linear and Multilinear Algebra. 2012;60:189–199].     

Toeplitz operators in n-dimensions

The interplay between the theory of Toeplitz operators on the circle and the theory of pseudodifferential operators on the line (i. e. Wiener-Hopf operators) is by now well-known and well-understood. In this article we show that there is a parallel situation in higher dimensions. To begin with, by using pseudodifferential multipliers, one can simplify the composition rules for Toeplitz operators, (§ 3), and describe precisely how Toeplitz operators of Bergmann type are related to Toeplitz operators of Szegö type (§ 9). Furthermore, it turns out that the ring of pseudodifferential operators on a compact manifold, M, is isomorphic with the ring of Toeplitz operators on an appropriate Grauert tube about M (§ § 4–6), and the ring of Weyl operators on ⁿ is isomorphic with the ring of Toeplitz operators on the complex ball in ⁿ (§ § 7–10).     

On Toeplitz localization operators

We present a unified approach to study properties of Toeplitz localization operators based on the Calderón and Gabor reproducing formula. We show that these operators with functional symbols on a plane domain may be viewed as certain pseudo-differential operators (with symbols on a line, or certain compound symbols).     

Extreme eigenvalues of real symmetric Toeplitz matrices

We exploit the even and odd spectrum of real symmetric Toeplitz matrices for the computation of their extreme eigenvalues, which are obtained as the solutions of spectral, or secular, equations. We also present a concise convergence analysis for a method to solve these spectral equations, along with an efficient stopping rule, an error analysis, and extensive numerical results.

     

Adjoints of slant Toeplitz operators

In this paper, we will use the Birkhoff's ergodic theorem to do some finer analysis on the spectral properties of slant Toeplitz operators. For example, we will show that if is an invertibleL function on the unit circle, then almost every point in (A ^* ) is not an eigenvalue ofA ^* . More specifically, we will show that the point spectrum ofA ^* is contained in a circle with positive radius.     

Symbols of truncated Toeplitz operators

We consider three topics connected with coinvariant subspaces of the backward shift operator in Hardy spaces Hp:

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properties of truncated Toeplitz operators;

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Carleson-type embedding theorems for the coinvariant subspaces;

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factorizations of pseudocontinuable functions from H¹.

These problems turn out to be closely connected and even, in a sense, equivalent. The new approach based on the factorizations allows us to answer a number of challenging questions about truncated Toeplitz operators posed by D. Sarason.     

Invariant subspaces of Toeplitz operators

This paper is devoted to the problem of the existence of invariant subspaces for Toeplitz operators. Let be a Lipschitzian arc in the plane and let f be a non-constant continuous functions on the unit circumference. It is proved that if there exists an open circle such that and if the modulus of continuity _f of the function f satisfies the condition then the Toeplitz operator T_f in the Hardy space H² has a nontrivial hyperinvariant subspace. For the proof of this theorem one makes use of the Lyubich-Matsaev theorem.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 126, pp. 170–179, 1983.I express my deep gratitude to E. M. Dyn'kin for useful discussions.     

Properties of generalized Toeplitz operators

An operatorX: is said to be a generalized Toeplitz operator with respect to given contractionsT ₁ andT ₂ ifX=T ₂XT₁ ^*. The purpose of this line of research, started by Douglas, Sz.-Nagy and Foia, and Pták and Vrbová, is to study which properties of classical Toeplitz operators depend on their characteristic relation. Following this spirit, we give appropriate extensions of a number of results about Toeplitz operators. Namely, Wintner's theorem of invertibility of analytic Toeplitz operators, Widom and Devinatz's invertibility criteria for Toeplitz operators with unitary symbols, Hartman and Wintner's theorem about Toeplitz operator having a Fredholm symbol, Hartman and Wintner's estimate of the norm of a compactly perturbed Toeplitz operator, and the non-existence of compact classical Toeplitz operators due to Brown and Halmos.Dedicated to our friend Cora Sadosky on the occasion of her sixtieth birthday     

Hyponormality of trigonometric Toeplitz operators

In this paper we establish a tractable and explicit criterion for the hyponormality of arbitrary trigonometric Toeplitz operators, i.e., Toeplitz operators with trigonometric polynomial symbols . Our criterion involves the zeros of an analytic polynomial induced by the Fourier coefficients of . Moreover the rank of the selfcommutator of is computed from the number of zeros of in the open unit disk and in counting multiplicity.

     

On operators commuting with Toeplitz operators modulo the compact operators

We prove that an operator on H² of the disc commutes modulo the compacts with all analytic Toeplitz operators if and only if it is a compact perturbation of a Toeplitz operator with symbol in H^∞ + C. Consequently, the essential commutant of the whole Toeplitz algebra is the algebra of Toeplitz operators with symbol in QC. The image in the Calkin algebra of the Toeplitz operators with symbol in H^∞ + C is a maximal abelian algebra. These results lead to a characterization of automorphisms of the algebra of compact perturbations of the analytic Toeplitz operators.     

Toeplitz operators on Bergman spaces

Let ? be an element in \(H^\infty (D) + C(\overline D )\) such that ?^* is locally sectorial. In this paper it is shown that the Toeplitz operator defined on the Bergman spaceA ² (D) is Fredholm. Also, it is proved that ifS is an operator onA ²(D) which commutes with the Toeplitz operatorT _? whose symbol ? is a finite Blaschke product, thenS H ^∞ (D) is contained inH ^∞ (D). Moreover, some spectral properties of Toeplitz operators are discussed, and it is shown that the spectrum of a class of Toeplitz operators defined on the Bergman spaceA ² (D), is not connected.