On the B-discrete spectrum

TL;DR

This paper introduces the B-discrete spectrum for unbounded operators, characterizes it via meromorphic resolvents, and explores its stability under perturbations, linking spectral properties to operator similarity.

Contribution

It defines the B-discrete spectrum for unbounded operators and establishes its equivalence with meromorphic resolvents, also analyzing spectral stability under perturbations.

Findings

A closed operator has a purely B-discrete spectrum iff it has a meromorphic resolvent.

The B-discrete spectrum is stable under certain perturbations.

Operators with quasisimilar paranormal inverses have identical spectra and B-discrete spectra.

Abstract

In this paper, we introduce the B-discrete spectrum of an unbounded closed operator and we prove that a closed operator has a purely B-discrete spectrum if and only if it has a meromorphic resolvent. After that, we study the stability of the B-discrete spectrum under several type of perturbations and we establish that two closed invertible linear operators having quasisimilar totally paranormal inverses have equal spectra and B-discrete spectra.