On New Approach to semi-Fredholm theory in unital C*-algebras

TL;DR

This paper extends a Hilbert-module approach to semi-Fredholm and semi-Weyl theory in unital C*-algebras, providing new proofs and broadening the algebraic framework for these operator theories.

Contribution

It introduces an extension of the Hilbert-module approach from Fredholm to semi-Fredholm and semi-Weyl theory, offering alternative proofs and unifying the theory.

Findings

Extended Hilbert-module approach to semi-Fredholm theory.

Provided new proofs for existing semi-Fredholm results.

Unified algebraic and Hilbert-module frameworks.

Abstract

Axiomatic Fredholm theory in unital C*-algebras was established by Keckic and Lazovic in [15]. Following the purely algebraic approach by Keckic and Lazovic, in [14] we extended further this theory to axiomatic semi-Fredholm and semi-Weyl theory in unital C*-algebras. However, recently, in [11] we developed another approach to axiomatic Fredholm theory in unital C*-algebras which is based on the theory of Hilbert modules and which is equivalent to the algebraic approach by Keckic and Lazovic. In this paper, we extend further this new Hilbert-module approach from Fredholm theory to semi-Fredholm and semi-Weyl theory in unital C*-algebras. Hence, we provide a new proof of the results in [14].