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On Left and Right Semi-B-Fredholm Operators

Alaa Hamdan, Mohammed Berkani

Abstract

To complete the study of Fredholm type operators of [10] and [11], we define in this paper the classes of left and right semi-B-Fredholm operators (Definition 3.1). Then, we prove that an operator $T \in L(X), X$ being a Banach space, is a left( resp. right) semi-B-Fredholm operator if and only if $T$ is the direct sum of left(resp. right) semi-Fredholm operator and a nilpotent one. This result extend the earlier characterization of B-Fredholm operators as the direct sum of a Fredholm operator and a nilpotent one obtained in [4,Theorem 2.7].

On Left and Right Semi-B-Fredholm Operators

Abstract

To complete the study of Fredholm type operators of [10] and [11], we define in this paper the classes of left and right semi-B-Fredholm operators (Definition 3.1). Then, we prove that an operator being a Banach space, is a left( resp. right) semi-B-Fredholm operator if and only if is the direct sum of left(resp. right) semi-Fredholm operator and a nilpotent one. This result extend the earlier characterization of B-Fredholm operators as the direct sum of a Fredholm operator and a nilpotent one obtained in [4,Theorem 2.7].
Paper Structure (3 sections, 8 theorems, 30 equations)

This paper contains 3 sections, 8 theorems, 30 equations.

Table of Contents

  1. Introduction
  2. Quasi-Fredholm Operators
  3. Left and Right semi-B-Fredholm operators

Key Result

Theorem 1.7

P7 Let $T \in L(X).$ Then T is a B-Fredholm operator if and only if there exists two closed subspaces $(M, N)$ in $Red(T)$ such that:

Theorems & Definitions (30)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Remark 1.6
  • Theorem 1.7
  • Definition 1.8
  • Definition 1.9
  • Lemma 1.10
  • ...and 20 more