A new approach in index theory

TL;DR

This paper introduces a new analytical index for continuous families of Fredholm operators parameterized by a topological space, extending classical index theory and proving related invariance and connectivity properties.

Contribution

It defines an extended index for families of Fredholm operators, proves its homotopy invariance, and explores topological properties of B-Fredholm operators.

Findings

Defined an analytical index for families of Fredholm operators.

Proved homotopy invariance of the new index.

Showed the space of B-Fredholm operators is path connected for separable Hilbert spaces.

Abstract

In this paper, we define an analytical index for a continuous family of Fredholm operators parameterized by a topological space $\mathbb{X}$ into a Hilbert space $H,$ as a sequence of integers, extending naturally the usual definition of the index and we prove the homotopy invariance of the index. We give also an extension of the Weyl theorem for normal continuous families and we prove that if $H$ is separable, then the space of B-Fredholm operators on $H$ is path connected.