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On Classes of Fredholm Type Operators

Alaa Hamdan, Mohammed Berkani

Abstract

Given an idempotent $p$ in a Banach algebra and following the study in \cite{P50} of p-invertibility, we consider here left p-invertibility, right p-invertibility and p-invertibility in the Calkin Algebra $\mathcal{C}(X),$ where $X$ is a Banach space. Then we define and study left and right generalized Drazin invertibility and we characterize left and right Drazin invertible elements in the Calkin algebra. Globally, this leads to define and characterize the classes of P-Fredholm, pseudo B-Fredholm and weak B-Fredholm operators.

On Classes of Fredholm Type Operators

Abstract

Given an idempotent in a Banach algebra and following the study in \cite{P50} of p-invertibility, we consider here left p-invertibility, right p-invertibility and p-invertibility in the Calkin Algebra where is a Banach space. Then we define and study left and right generalized Drazin invertibility and we characterize left and right Drazin invertible elements in the Calkin algebra. Globally, this leads to define and characterize the classes of P-Fredholm, pseudo B-Fredholm and weak B-Fredholm operators.
Paper Structure (4 sections, 9 theorems, 22 equations)

This paper contains 4 sections, 9 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. On P-Fredholm Operators
  3. On Pseudo B-Fredholm Operators
  4. On Weak B-Fredholm Operators

Key Result

Theorem 2.2

Let $T \in L(X).$ Then there exist an idempotent $p=\Pi(P) \in \mathcal{C}(X)$ such that $T$ is a left semi-P-Fredholm operator if and only if there exists an element $S \in L(X)$ such that $STS-S, S^2T-S, TST-ST^2$ are compact, there exists $V \in L(X)$ such that the commutator $[V, ST]$ and $V(I+T

Theorems & Definitions (32)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Remark 1.6
  • Definition 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • ...and 22 more