Left multipliers of reproducing kernel Hilbert C⁎-modules and the Papadakis theorem

Abstract

We give a modified definition of a reproducing kernel Hilbert C⁎-module (shortly, RKHC⁎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 RKHC⁎Ms and find their reproducing kernel. In addition, we deal with left multipliers of RKHC⁎Ms. Under some mild conditions, it is shown that one can make a new RKHC⁎M via a left multiplier. Moreover, we introduce the Berezin transform of an operator in the context of RKHC⁎Ms and construct a unital subalgebra of the unital C⁎-algebra consisting of adjointable maps on an RKHC⁎M and show that it is closed with respect to a certain topology. Finally, the Papadakis theorem is extended to the setting of RKHC⁎M, and in order for the multiplication of two specific functions to be in the Papadakis RKHC⁎M, some conditions are explored.