Stability Theory for Nullity and Deficiency of Linear Relations

TL;DR

This paper investigates the stability of nullity and deficiency of closed linear relations between Banach spaces under perturbations, establishing conditions for their invariance within a certain complex disk.

Contribution

It introduces a stability framework for nullity and deficiency of linear relations under perturbations, extending the understanding of their behavior in Banach space settings.

Findings

Existence of a constant >0 ensuring stability of nullity and deficiency

Stability holds for all perturbation parameters  with ||<

Provides conditions for invariance of linear relation properties under perturbations

Abstract

Let $\mathcal A$ and $\mathcal B$ be two closed linear relation acting between two Banach spaces $X$ and $Y$ and let $\lambda$ be a complex number. We study the stability of the nullity and deficiency of $\mathcal A$ when it is perturbed by $\lambda\mathcal B$. In particular, we show the existence of a constant $\rho>0$ for which both the nullity and deficiency of $\mathcal A$ remain stable under perturbation by $\lambda\mathcal B$ for all $\lambda$ inside the disk $\vert \lambda\vert <\rho$.