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Paracommutators. Brief introduction, open problems.

Jaak Peetre (1989)

Revista Matemática de la Universidad Complutense de Madrid

We review the basic facts about the theory of paracommutators in Rn (sec S. Janson, J. Peetre, Trans. Am. Math. Soc. 305 (1988), 467504). We also give an interpretation of paracommutators from the point of view of group representations. This suggests a generalization to more general groups. Here we sketch a theory of paracommutators over stratified groups. This include the famous Heisenberg group. Finally, we take up the question of generalizing the notion of Schatten-von Neumann trace ideals to...

Perturbed Toeplitz operators and radial determinantal processes

Torsten Ehrhardt, Brian Rider (2013)

Annales de l'I.H.P. Probabilités et statistiques

We study a class of rotation invariant determinantal ensembles in the complex plane; examples include the eigenvalues of Gaussian random matrices and the roots of certain families of random polynomials. The main result is a criterion for a central limit theorem to hold for angular statistics of the points. The proof exploits an exact formula relating the generating function of such statistics to the determinant of a perturbed Toeplitz matrix.

Podal subspaces on the unit polydisk

Kunyu Guo (2002)

Studia Mathematica

Beurling's classical theorem gives a complete characterization of all invariant subspaces in the Hardy space H²(D). To generalize the theorem to higher dimensions, one is naturally led to determining the structure of each unitary equivalence (resp. similarity) class. This, in turn, requires finding podal (resp. s-podal) points in unitary (resp. similarity) orbits. In this note, we find that H-outer (resp. G-outer) functions play an important role in finding podal (resp. s-podal) points. By the methods...

Positive Schatten class Toeplitz operators on the ball

Boo Rim Choe, Hyungwoon Koo, Young Joo Lee (2008)

Studia Mathematica

On the harmonic Bergman space of the ball, we give characterizations for an arbitrary positive Toeplitz operator to be a Schatten class operator in terms of averaging functions and Berezin transforms.

Positive Toeplitz operators between the pluriharmonic Bergman spaces

Eun Sun Choi (2008)

Czechoslovak Mathematical Journal

We study Toeplitz operators between the pluriharmonic Bergman spaces for positive symbols on the ball. We give characterizations of bounded and compact Toeplitz operators taking a pluriharmonic Bergman space b p into another b q for 1 < p , q < ∞ in terms of certain Carleson and vanishing Carleson measures.

Preconditioners and Korovkin-type theorems for infinite-dimensional bounded linear operators via completely positive maps

K. Kumar, M. N. N. Namboodiri, S. Serra-Capizzano (2013)

Studia Mathematica

The classical as well as noncommutative Korovkin-type theorems deal with the convergence of positive linear maps with respect to different modes of convergence, like norm or weak operator convergence etc. In this article, new versions of Korovkin-type theorems are proved using the notions of convergence induced by strong, weak and uniform eigenvalue clustering of matrix sequences with growing order. Such modes of convergence were originally considered for the special case of Toeplitz matrices and...

Problème spectral inverse et équation de Szegö cubique

Patrick Gérard, Sandrine Grellier (2011/2012)

Séminaire Laurent Schwartz — EDP et applications

Dans un exposé précédent [1], nous avons justifié l’introduction de l’équation de Szegö cubique comme cas modèle d’équation de type Schrödinger sans dispersion. Ce cas modèle s’est révélé être intéressant sous divers aspects [2]. Dans cet exposé, nous nous attacherons à montrer comment la complète intégrabilité de l’équation de Szegö cubique permet de résoudre un problème spectral inverse pour les opérateurs de Hankel.

Product equivalence of quasihomogeneous Toeplitz operators on the harmonic Bergman space

Xing-Tang Dong, Ze-Hua Zhou (2013)

Studia Mathematica

We present here a quite unexpected result: If the product of two quasihomogeneous Toeplitz operators T f T g on the harmonic Bergman space is equal to a Toeplitz operator T h , then the product T g T f is also the Toeplitz operator T h , and hence T f commutes with T g . From this we give necessary and sufficient conditions for the product of two Toeplitz operators, one quasihomogeneous and the other monomial, to be a Toeplitz operator.

Products of Toeplitz operators and Hankel operators

Yufeng Lu, Linghui Kong (2014)

Studia Mathematica

We first determine when the sum of products of Hankel and Toeplitz operators is equal to zero; then we characterize when the product of a Toeplitz operator and a Hankel operator is a compact perturbation of a Hankel operator or a Toeplitz operator and when it is a finite rank perturbation of a Toeplitz operator.

Projections onto the spaces of Toeplitz operators

Marek Ptak (2005)

Annales Polonici Mathematici

Projections onto the spaces of all Toeplitz operators on the N-torus and the unit sphere are constructed. The constructions are also extended to generalized Toeplitz operators and applied to show hyperreflexivity results.

Properties of two variables Toeplitz type operators

Elżbieta Król-Klimkowska, Marek Ptak (2016)

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

The investigation of properties of generalized Toeplitz operators with respect to the pairs of doubly commuting contractions (the abstract analogue of classical two variable Toeplitz operators) is proceeded. We especially concentrate on the condition of existence such a non-zero operator. There are also presented conditions of analyticity of such an operator.

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