Characterization results and Riesz spectral theory for compact linear relations

Abstract

This paper extends Riesz’s theory of compact operators to the general setting of compact linear relations. Using compact selections and studying invariant subspaces, we develop new characterizations of compactness in the multivalued setting. In doing so, we lay the groundwork for a spectral theory of compact linear relations. Our investigation encompasses key structural  properties, including the behavior of the range,  Aronszajn–Smith and Riesz–Schauder type results, and spectral notions such as ascent, descent, isolated spectral points, the analytic core, and the quasinilpotent part. The results highlight both deep analogies and essential differences between single-valued compact operators and their multivalued counterparts.