We show that for any localization operator on the Fock space with polynomial window, there exists a constant coefficient linear partial differential operator D such that the localization operator with symbol f coincides with the Toeplitz operator with symbol Df . An analogous result also holds in the context of Bergman spaces on bounded symmetric domains. This verifies a recent conjecture of Co...

We present forms of the classical Riesz–Kolmogorov theorem for compactness that are applicable in a wide variety settings. In particular, our theorems apply to classify precompact subsets Lebesgue space $$L^2$$ , Paley–Wiener spaces, weighted Bargmann–Fock and scale Besov–Sobolev spaces holomorphic functions includes Bergman general domains as well Hardy Dirichlet space. criteria characterize c...

whenever Φ is such that this extends to a bounded operator on Lpr0, asq. Whenever a is of no importance we will omit it from the notation. We abbreviate by saying that WΦ is a TWH-operator. These operators (or rather, unitarily equivalent ones) also go under the name finite interval convolution operators, truncated Hankel operators or Toeplitz operators on the Paley-Wiener space. See e.g. [1, 2...

By a famous result, functions in backward shift invariant subspaces in Hardy spaces are characterized by the fact that they admit a pseudocontinuation a.e. on T. More can be said if the spectrum of the associated inner function has holes on T. Then the functions of the invariant subspaces even extend analytically through these holes. We will discuss the situation in weighted backward shift inva...

For truncated Toeplitz operators, which are compressions of multiplication operators to model subspaces of the Hardy space H2 , we obtain criteria for commutation relations. The results show an analogy to the case of Toeplitz matrices, and they extend the theory of Sedlock algebras. Mathematics subject classification (2010): 47B32, 47B35, 47B37.

The paper is devoted to the study of Toeplitz operators with piecewise continuous symbols. We clarify the geometric regularities of the behaviour of the essential spectrum of Toeplitz operators in dependence on their crucial data: the angles between jump curves of symbols at a boundary point of discontinuity and on the limit values reached by a symbol at that boundary point. We show then that t...

Compressions of Toeplitz operators to coinvariant subspaces of H are called truncated Toeplitz operators. We study two questions related to these operators. The first, raised by Sarason, is whether boundedness of the operator implies the existence of a bounded symbol; the second is the Reproducing Kernel Thesis. We show that in general the answer to the first question is negative, and we exhibi...