Linear and non-linear functions on quasi-normed spaces

Abstract

This thesis develops operator theory in quasi-Banach spaces, focusing on three areas. First, we extend fundamental theorems ( Banach Steinhaus, closed graph, etc) and we develop continuity and boundedness theory for multilinear operators. Second, to multivalued linear relations. Third, we study nonlinear operators in p-normed spaces, particularly Lipschitz operators, and characterize two-Lipschitz left-hand quotient operator ideals A−1 ◦ ABLip, identifying important classes such as Grothendieck and Rosenthal operators. Our work bridges linear and nonlinear operator theory.