Dynamics of Properties of Toeplitz Operators with Radial Symbols

Abstract

In the case of radial symbols we study the behavior of different properties (boundedness, compactness, spectral properties, etc.) of Toeplitz operators T (λ) a acting on weighted Bergman spaces A 2 λ (D) over the unit disk D, in dependence on λ, and compare their limit behavior under λ → +∞ with corresponding properties of the initial symbol a. Mathematics Subject Classification (2000). Primary 47B35; Secondary 47A25, 47B10, 46E22.

Mixed norm spaces of analytic functions as spaces of generalized fractional derivatives of functions in Hardy type spaces

“…These conditions were obtained (see, for instance, [14]) in terms of behavior of some means (averages) of the symbol a when r → 1. There also many examples of badly behaved oscillating and unbounded symbols that generate even compact operators.…”

Dynamics of Properties of Toeplitz Operators on the Upper Half-Plane: Parabolic Case

“…We mention a certain difference between the already studied elliptic case [8] and the remaining cases. In particular, in the elliptic case Toeplitz operators have a discrete spectrum and can be compact even for symbols that are unbounded near the boundary, while in both parabolic and hyperbolic cases, Toeplitz operators always have only a continuous spectrum and, being nonzero, can not be compact.…”

“…In the present paper we consider the model case for parabolic pencils, while the other paper, [9] from this two-paper set is devoted to the study of the model case for hyperbolic pencils. Together, these papers cover the material that remained uncovered after [8]. The results for other (non model) cases can be easily obtained by means of Möbius transformations.…”

“…Both of them are motivated by the same ideas and are a continuation of our research started in [8]. We mentioned in [8] the papers [3], [4], [5], [12], [13], where Toeplitz operators with smooth (or continuous) symbols acting on the weighted Bergman spaces, as well as C * -algebras generated by such operators, naturally appear in the context of problems in mathematical physics. In particular, recall that given a smooth symbol a = a(z), the family of Toeplitz operators T a = {T (h) a }, with h ∈ (0, 1), is considered under the Berezin quantization procedure ( [3], [4]).…”

“…The model case for elliptic pencils, Toeplitz operators on the unit disk with radial symbols, has been considered in [8]. In the present paper we consider the model case for parabolic pencils, while the other paper, [9] from this two-paper set is devoted to the study of the model case for hyperbolic pencils.…”