Projectively invariant Hilbert-Schmidt kernels and convolution type operators

Abstract

We consider positive definite kernels which are invariant under a multiplier and an action of a semigroup with involution, and construct the associated projective isometric representation on a Hilbert C*-module. We introduce the notion of C*-valued Hilbert Schmidt kernels associated with two sequences and construct the corresponding reproducing Hilbert C*-module. We also discuss projective invariance of Hilbert-Schmidt kernels. We prove that the range of a convolution type operator associated with a Hilbert-Schmidt kernel coincides with the reproducing Hilbert C*-module associated with its convolution kernel. We show that the integral operator associated with a Hilbert-Schmidt kernel is Hilbert-Schmidt. Finally, we discuss a relation between an integral type operator and convolution type operator.