JEDNOSTKA NAUKOWA KATEGORII A+

# Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

## Projectively invariant Hilbert–Schmidt kernels and convolution type operators

### Tom 213 / 2012

Studia Mathematica 213 (2012), 61-79 MSC: Primary 46E22, 46C50; Secondary 46L08, 46N55. DOI: 10.4064/sm213-1-5

#### Streszczenie

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.

#### Autorzy

• Jaeseong HeoDepartment of Mathematics
Research Institute for Natural Sciences
Hanyang University
Seoul 133-791, Korea
e-mail

## Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Odśwież obrazek