On the generalized resolvent of linear pencils in Banach spaces

TL;DR

This paper investigates the existence and explicit form of generalized resolvents for linear pencils in Banach spaces, providing practical criteria and applications to various classes of operators, thereby extending existing theoretical results.

Contribution

It offers new criteria and explicit formulas for generalized resolvents of linear pencils in Banach spaces, extending previous results and applying to specific operator classes.

Findings

Provided practical criteria for existence of generalized resolvents.

Derived explicit expressions for the generalized resolvent.

Extended results to finite rank, Fredholm, and semi-Fredholm operators.

Abstract

Utilizing the stability characterizations of generalized inverses of linear operator, we investigate the existence of generalized resolvents of linear pencils in Banach spaces. Some practical criterions for the existence of generalized resolvents of the linear pencil $\lambda\rightarrow T-\lambda S$ are provided and an explicit expression of the generalized resolvent is given. As applications, the characterization for the Moore-Penrose inverse of the linear pencil to be its generalized resolvent and the existence of the generalized resolvents of linear pencils of finite rank operators, Fredholm operators and semi-Fredholm operators are also considered. The results obtained in this paper extend and improve many results in this area.