Strongly ϕ-summing operators

Abstract

The primary objective of this master thesis is to systematically investigate the geometric and structural properties of specialized classes of summation mappings, tracking the theoretical transition from the classical linear framework to non-linear domain structures. We begin by reviewing the algebraic and topological foundations of Modulus, Young, and s-Young functions, which establish the architectural setup for generalized Orlicz and Calderón-Lozanovskii sequence spaces, alongside pointed Lipschitz spaces and their canonical linearizations via Lipschitz-free spaces. We then analyze the classical theory of bounded linear ϕ-summing operators acting between Banach spaces, revisiting the foundational Pietsch Domination and Factorization Theorems via regular Borel probability measures. Moving into the non-linear domain, we explore the topological completeness and ideal properties of the metric linear space of Lipschitz ϕ-summing operators Πϕ(E, F) under subadditive modulus functions. Finally, we establish the core contribution of this research by introducing and thoroughly investigating the class of Strongly φ-Summing Operators within the framework of general Orlicz functions. By employing Ky Fan’s subconvex separation lemma, we formulate and prove the Main Domination Theorem, demonstrating that these generalized operators admit a deep structural characterization through Borel probability measures and infimal convolutions (φ1 ⊕ φ2).