Boundedness and closedness of linear relations

TL;DR

This paper investigates the properties of linear relations, including boundedness and closedness, introduces a new induced linear operator, and extends classical results to multi-valued operators in Banach spaces.

Contribution

It introduces a new linear operator induced by linear relations and generalizes classical operator results to multi-valued linear relations.

Findings

Characterizations of closedness, closability, and boundedness of linear relations.

Completion of the closed graph theorem for linear relations in Banach spaces.

Analysis of stability of closedness under perturbations.

Abstract

This paper studies boundedness and closedness of linear relations, which include both single-valued and multi-valued linear operators. A new (single-valued) linear operator induced by a linear relation is introduced, and its relationships with other two important induced linear operators are established. Several characterizations for closedness, closability, bundedness, relative boundedness, and boundedness from below (above) of linear relations are given in terms of their induced linear operators. In particular, the closed graph theorem for linear relations in Banach spaces is completed, and stability of closedness of linear relations under bounded and relatively bounded perturbations is studied. The results obtained in the present paper generalize the corresponding results for single-valued linear operators to multi-valued linear operators, and some improve or relax certain assumptions of the related existing results.