Left multipliers of reproducing kernel Hilbert $C^{\ast }$-modules and the Papadakis theorem

Abstract

We give a modified definition of a reproducing kernel Hilbert $C^{\ast }$-module (shortly, $RKH{C}^{\ast }M$) without using the condition of self-duality and discuss some related aspects; in particular, an interpolation theorem is presented. We investigate the exterior tensor product of $RKH{C}^{\ast }M$s and find their reproducing kernel. In addition, we deal with left multipliers of $RKH{C}^{\ast }M$s. Under some mild conditions, it is shown that one can make a new $RKH{C}^{\ast }M$ via a left multiplier. Moreover, we introduce the Berezin transform of an operator in the context of $RKH{C}^{\ast }M$s and construct a unital subalgebra of the unital $C^{\ast }$-algebra consisting of adjointable maps on an $RKH{C}^{\ast }M$ and show that it is closed with respect to a certain topology. Finally, the Papadakis theorem is extended to the setting of $RKH{C}^{\ast }M$, and in order for the multiplication of two specific functions to be in the Papadakis $RKH{C}^{\ast }M$, some conditions are explored.

15 pages