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Fredholm-type Operators and Index

Alaa Hamdan, Mohammed Berkani

TL;DR

This work extends Fredholm-type operator theory by developing P-Fredholm and pseudo-B-Fredholm classes through the quotient algebra \(\mathcal{C}_0(X)=L(X)/F_0(X)\) via the homomorphism \(\pi\). It constructs lifting of idempotents from \(\mathcal{C}_0(X)\) to \(L(X)\), relates \(\pi(T)\)–invertibility to semi-Fredholm properties of \(T\), and defines generalized Drazin invertibility and quasinilpotence in \(\mathcal{C}_0(X)\). A central contribution is the introduction of an index ind\((T)\) for pseudo-B-Fredholm operators, defined as the index of an associated operator \(T+P\) and shown to be well-defined, stable under compact perturbations, and to reduce to the classical Fredholm index on standard semi-Fredholm cases. The results yield a robust framework for index theory beyond Fredholm operators, including stability and perturbation properties, and connect the Calkin-algebra viewpoint with quotient-algebra methods. This broadens the applicability of index theory to a wider class of Fredholm-type operators and clarifies the role of associated idempotents in the Calkin and quotient algebras.

Abstract

While in \cite{HB} we studied classes of Fredholm-type operators defined by the homomorphism $Π$ from $L(X)$ onto the Calkin algebra $\mathcal{C}(X)$, $X$ being a Banach space, we study in this paper two classes of Fredholm-type operators defined by the homomorphism $π$ from $L(X)$ onto the algebra $\mathcal{C}_0(X)= L(X)/F_0(X),$ where $F_0(X)$ is the ideal of finite rank operators in $L(X).$ Then we define an index for Fredholm-type operators and we show that this new index satisfies similar properties as the usual Fredholm index.

Fredholm-type Operators and Index

TL;DR

This work extends Fredholm-type operator theory by developing P-Fredholm and pseudo-B-Fredholm classes through the quotient algebra \(\mathcal{C}_0(X)=L(X)/F_0(X)\) via the homomorphism . It constructs lifting of idempotents from \(\mathcal{C}_0(X)\) to \(L(X)\), relates \(\pi(T)\)–invertibility to semi-Fredholm properties of , and defines generalized Drazin invertibility and quasinilpotence in \(\mathcal{C}_0(X)\). A central contribution is the introduction of an index ind\((T)\) for pseudo-B-Fredholm operators, defined as the index of an associated operator and shown to be well-defined, stable under compact perturbations, and to reduce to the classical Fredholm index on standard semi-Fredholm cases. The results yield a robust framework for index theory beyond Fredholm operators, including stability and perturbation properties, and connect the Calkin-algebra viewpoint with quotient-algebra methods. This broadens the applicability of index theory to a wider class of Fredholm-type operators and clarifies the role of associated idempotents in the Calkin and quotient algebras.

Abstract

While in \cite{HB} we studied classes of Fredholm-type operators defined by the homomorphism from onto the Calkin algebra , being a Banach space, we study in this paper two classes of Fredholm-type operators defined by the homomorphism from onto the algebra where is the ideal of finite rank operators in Then we define an index for Fredholm-type operators and we show that this new index satisfies similar properties as the usual Fredholm index.
Paper Structure (4 sections, 18 theorems, 13 equations)

This paper contains 4 sections, 18 theorems, 13 equations.

Table of Contents

  1. Introduction
  2. P-Fredholm Operators through $\mathcal{C}_0(X)$
  3. Pseudo B-Fredholm Operators through $\mathcal{C}_0(X).$
  4. Index of Fredholm type Operators

Key Result

Theorem 2.1

Let $p_0$ be an idempotent element of the algebra $\mathcal{C}_0(X).$ Then there exists an idempotent $P \in L(X)$ such that $\pi(P)= p_0.$

Theorems & Definitions (52)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Remark 1.7
  • Theorem 2.1
  • proof
  • Theorem 2.2
  • ...and 42 more