The Essential Norm of Operators on $${A^p_\alpha(\mathbb{B}_n)}$$

Abstract

In this paper we characterize the compact operators on A p α (B n ) when 1 < p < ∞ and α > −1. The main result shows that an operator on A p α (B n ) is compact if and only if it belongs to the Toeplitz algebra and its Berezin transform vanishes on the boundary of the ball.

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

“…A version of this result in the classical Bergman space setting was first proved by Suárez in [31]. Related results were later given in [4,20,22,25]. …”

“…For the proof see [21]. Related results can be found in [4,8,22,31] where it is shown that nice domains, such as the unit ball, polydisc, or C n have this property.…”

“…This is not a trivial assumption and it is (at this point) not known whether this holds for all Bergman-type spaces (see [21]). It does hold in the standard Bergman spaces on the ball and polydisc and also on the Fock space see [4,20,22,31]. Thus, the following theorem can be viewed as an extension of the main theorems in [4,20,22,31] to the ℓ 2 -valued setting.…”

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

Compact Operators on the Bergman Spaces with Variable Exponents on the Unit Disc of C

“…The behaviour of these operators on the Hardy spaces, Bergman spaces, and Fock spaces has been studied widely and a lot of results are available in the literature. The characterization of compactness has been studied in [2][3][4][5][6][7][8] just to cite a few. Given Ω ⊂ R , a measurable function : Ω → [1, ∞) will be called a variable exponent.…”