# IJPAM: Volume 23, No. 2 (2005)

**ON THE CONSTRUCTION OF OPERATOR RANGE**

TOPOLOGY AND TOPOLOGICAL UNIFORM DESCENT OF

KATO OPERATORS IN BANACH SPACES

TOPOLOGY AND TOPOLOGICAL UNIFORM DESCENT OF

KATO OPERATORS IN BANACH SPACES

Department of Mathematics

Faculty of Science

Mohamed First University

Oujda, MOROCCO

e-mail: lahrech@sciences.univ-oujda.ac.ma

e-mail: ouahab@sciences.univ-oujda.ac.ma

e-mail: benbrik@sciences.univ-oujda.ac.ma

e-mail: mbarki@sciences.univ-oujda.ac.ma

**Abstract.**We consider a Kato operator on a Banach space (i.e., a
closed operator satisfying the following condition: the null
space of has a topological complement invariant by
) such that for every positive integer
, where denotes the range of .
We first prove that given a Kato operator , we can always give
an operator range topology on under which it becomes a
Banach space continuously embedded in for every positive
integer . So, in our result, to construct the range topology of
we only require that
is closed, but not which can be not closed.

Using this result, we study the structure of the class of Kato
operators. Our study focuses on the sequences of ranges
.

**Received: **June 13, 2005

**AMS Subject Classification: **47A10, 47A53

**Key Words and Phrases: **Kato operators, operator range topology

**Source:** International Journal of Pure and Applied Mathematics

**ISSN:** 1311-8080

**Year:** 2005

**Volume:** 23

**Issue:** 2