IJPAM: Volume 23, No. 2 (2005)

ON THE CONSTRUCTION OF OPERATOR RANGE
TOPOLOGY AND TOPOLOGICAL UNIFORM DESCENT OF
KATO OPERATORS IN BANACH SPACES

S. Lahrech$^1$, A. Ouahab$^2$, A. Benbrik$^3$, A. Mbarki$^4$
$^{1,2,3,4}$Department of Mathematics
Faculty of Science
Mohamed First University
Oujda, MOROCCO
$^1$e-mail: lahrech@sciences.univ-oujda.ac.ma
$^2$e-mail: ouahab@sciences.univ-oujda.ac.ma
$^3$e-mail: benbrik@sciences.univ-oujda.ac.ma
$^4$e-mail: mbarki@sciences.univ-oujda.ac.ma


Abstract.We consider a Kato operator $A$ on a Banach space $X$ (i.e., a closed operator $A$ satisfying the following condition: the null space $N(A)$ of $A$ has a topological complement $L$ invariant by $A$) such that for every positive integer $n$  $A^n({\cal D}(A^{n+1}))=R(A^n)$, where $R(A^n)$ denotes the range of $A^n$. We first prove that given a Kato operator $A$, we can always give an operator range topology on $R(A^n)$ under which it becomes a Banach space continuously embedded in $X$ for every positive integer $n$. So, in our result, to construct the range topology of $A^n$ we only require that $A$ is closed, but not $A^n$ 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 $\{R(A^n)\}$.

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