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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm213-1-5,
author = {Jaeseong Heo},
title = {Projectively invariant Hilbert-Schmidt kernels and convolution type operators},
journal = {Studia Mathematica},
volume = {209},
year = {2012},
pages = {61-79},
zbl = {1272.46045},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm213-1-5}
}
Jaeseong Heo. Projectively invariant Hilbert-Schmidt kernels and convolution type operators. Studia Mathematica, Tome 209 (2012) pp. 61-79. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm213-1-5/