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

Abstract

We study the compactness of some classes of bounded operators on the Bergman space with variable exponent. We show that via extrapolation, some results on boundedness of the Toeplitz operators with general 1 symbols and compactness of bounded operators on the Bergman spaces with constant exponents can readily be extended to the variable exponent setting. In particular, if is a finite sum of finite products of Toeplitz operators with symbols from class , then is compact if and only if the Berezin transform of vanishes on the boundary of the unit disc.