A reproducing kernel thesis for operators on Bergman-type function spaces

Abstract

In this paper we consider the reproducing kernel thesis for boundedness and compactness for various operators on Bergman-type spaces. In particular, the results in this paper apply to the weighted Bergman space on the unit ball, and more generally to weighted Fock spaces.2000 Mathematics Subject Classification. 32A36, 32A, 47B05, 47B35.

The Essential Norm of Operators on $$\ell ^{2}$$ ℓ 2 -Valued Bergman-Type Function Spaces

“…However, results of this type are well known for the classical Bergman spaces on the ball. See for example [4,8,21,22,31].…”

“…For a more detailed discussion of examples of Bergman-type spaces, see [21]. Clearly, any Bergmantype space can be extended to a ℓ 2 -valued Bergman space and so the ℓ 2 -valued versions of these spaces are ℓ 2 -valued Bergman-type spaces.…”

“…In [21], Mitkovski and Wick show that in a wide variety of classical functions spaces (they call these spaces Bergman-type function spaces), many properties of an operator can be determined by studying its behavior on the normailzed reproducing kernels. Thus, their results are "Reproducing Kernel Thesis" (RKT) statements.…”

“…The unified approach developed in [21] was used to solve two types of problems relating to operators on classical function spaces: boundedness and compactness. The goal of this paper is to extend this approach to the case of ℓ 2 -valued Bergman type function spaces and to prove results relating to boundedness and compactness of operators for a general class of ℓ 2 -valued Bergman type function spaces.…”

“…The goal of this paper is to extend this approach to the case of ℓ 2 -valued Bergman type function spaces and to prove results relating to boundedness and compactness of operators for a general class of ℓ 2 -valued Bergman type function spaces. The proofs in this paper are essentially the same as the corresponding proofs from [21]. The only adjustments are that our integrals are now vector-valued and we must use a version of the classical Schur's test for integral operators with matrix-valued kernels.…”