Projectively invariant Hilbert–Schmidt kernels and convolution type operators

Jaeseong Heo¹

¹ Department of Mathematics Research Institute for Natural Sciences Hanyang University Seoul 133-791, Korea

Studia Mathematica, Tome 213 (2012) no. 1, pp. 61-79

Résumé

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.

Détail

DOI : 10.4064/sm213-1-5

Keywords: consider positive definite kernels which invariant under multiplier action semigroup involution construct associated projective isometric representation hilbert * module introduce notion * valued hilbert schmidt kernels associated sequences construct corresponding reproducing hilbert * module discuss projective invariance hilbert schmidt kernels prove range convolution type operator associated hilbert schmidt kernel coincides reproducing hilbert * module associated its convolution kernel integral operator associated hilbert schmidt kernel hilbert schmidt finally discuss relation between integral type operator convolution type operator