Propertiesandfor Bounded Linear Operators

M H M Rashid

doi:10.1155/2013/848176

Abstract

We shall consider properties which are related to Weyl type theorem for bounded linear operators , defined on a complex Banach space . These properties, that we callproperty, means that the set of all poles of the resolvent of of finite rank in the usual spectrum are exactly those points of the spectrum for which is an upper semi-Fredholm with index less than or equal to 0 and we callproperty, means that the set of all poles of the resolvent of in the usual spectrum are exactly those points of the spectrum for which is an upper semi--Fredholm with index less than or equal to 0. Properties and are related to a strong variants of classical Weyl’s theorem, the so-called property and property We shall characterize properties and in several ways and we shall also describe the relationships of it with the other variants of Weyl type theorems. Our main tool is localized version of the single valued extension property. Also, we consider the properties and in the frame of polaroid type operators.

Highlights

Introduction andPreliminary roughout this paper, XX denotes an in nite-dimensional complex Banach space, L(XXX the algebra of all bounded linear operators on XX
For an operator TT T TTTTT we shall denote by αααααα the dimension of the kernel ker(TTT, and by ββββββ the codimension of the range R(TTT
Be the class of all lower semi-Fredholm operators. e class of all semi-Fredholm operators is de ned by SF±(XXXXX SF+(XXX X SF−(XXX, while the class of all Fredholm operators is de ned by SF(XXXXX SF+(XXX X SF−(XXX

Summary

Introduction

Introduction andPreliminary roughout this paper, XX denotes an in nite-dimensional complex Banach space, L(XXX the algebra of all bounded linear operators on XX. From eorem 2.8 of [4] it follows that if TT T TTTTT is le Drazin invertible, TT is an upper semi-BB-Fredholm operator of index less than or equal to 0.

Results

Conclusion