Index of continuous families of bounded linear operators in Banach spaces and application

TL;DR

This paper introduces an analytical index for continuous families of Fredholm operators in Banach spaces, explores the Weyl spectrum's properties, and investigates its continuity, contributing to operator theory and spectral analysis.

Contribution

It defines a new analytical index for parameterized Fredholm operators and studies the continuity of the Weyl spectrum in Banach spaces.

Findings

Defined an analytical index for continuous families of Fredholm operators

Analyzed the Weyl spectrum's continuity properties

Extended spectral theory in the context of Banach spaces

Abstract

In this paper, we define an analytical index for continuous families of Fredholm operators parameterized by a topological space $\mathbb{X}$ into a Banach space $X.$ We also consider the Weyl spectrum for continuous families of bounded linear operators and we study its continuity.