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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.