In this paper, we focus on the weighted Bergman spaces Aφp in D with φ∈W0. We first give characterizations of those finite positive Borel measures µ such that embedding Aφp⊂Lμq is bounded or compact for 0<p,q<∞. Then describe Toeplitz operators Tμ from one space to another Aφq all possible Finally, characterize Schatten class Aφ2.

In this paper, the Fredholm properties of some Toeplitz operators on Dirichlet spaces be discussed, and the essential spectra of Toeplitz operators with symbols in C 1 or H ∞ 1 + C 1 be computed.

In this paper we prove that if S equals a finite sum of finite products of Toeplitz operators on the Bergman space of the unit disk, then S is compact if and only if the Berezin transform of S equals 0 on ∂D. This result is new even when S equals a single Toeplitz operator. Our main result can be used to prove, via a unified approach, several previously known results about compact Toeplitz oper...

These are the notes of the lecture course given at LTCC in 2015. The aim of the course is to consider the following three classes of operators: Toeplitz and Hankel operators on the Hardy space on the unit circle and Toeplitz operators on the Bergman space on the unit disk. For each of these three classes of operators, we consider the following questions: boundedness and estimates or explicit ex...

A trace formula for Toeplitz operators was proved by Boutet de Monvel and Guillemin in the setting of general Toeplitz structures. Here we give a local version of this result for a class of Toeplitz operators related to continuous groups of symmetries on quantizable compact symplectic manifolds. The local trace formula involves certain scaling asymptotics along the clean fixed locus of the Hami...

Recently the authors characterized Fredholmn properties of Toeplitz operators on weighted Fock spaces when Laplacian weight function is bounded below and above. In present work extend their characterization to doubling with a subharmonic whose measure. The geometry induced by Bergman metric for much more complicated than that Euclidean used in all previous cases study Fredholmness, which leads ...