Samuel multiplicities and Browder Spectrum of Operator Matrices

TL;DR

This paper explores the relationships between classes of semi-Fredholm operators, Samuel multiplicities, and Browder spectrum of operator matrices, providing characterizations of spectral sets for upper triangular operator matrices.

Contribution

It establishes the equivalence of certain semi-Fredholm operator classes and characterizes spectral sets of operator matrices using Samuel multiplicities.

Findings

Equivalence of semi-Fredholm operator classes with known classes

Characterization of spectral sets of operator matrices

Application of Samuel multiplicities to spectral analysis

Abstract

we show that the definitions of some classes of semi-Fredholm operators, which use the language of algebra and first introduced by X. Fang in [8], are equivalent to that of some well-known operator classes. For example, the concept of shift-like semi-Fredholm operator on Hilbert space coincide with that of upper semi-Browder operator. For applications of Samuel multiplicities we characterize the sets of $\bigcap_{C\in B(K,\,H)}\sigma_{ab}(M_{C}),\bigcap_{C\in B(K,\,H)}\sigma_{sb}(M_{C})$ and $\bigcap_{C\in B(K,\,H)}\sigma_{b}(M_{C}),$ respectively, where $M_{C}=({array}{cc}A&C 0&B {array})$ denotes a 2-by-2 upper triangular operator matrix acting on the Hilbert space $H\oplus K$.