Operators between Hilbert Spaces Viewed as Only Linear Topological -- Towards a Classification

TL;DR

This paper investigates the classification of bounded linear operators between Hilbert spaces based solely on their linear topological properties, aiming to identify invariants that distinguish their equivalence classes.

Contribution

It introduces a framework for classifying operators between Hilbert spaces using topological invariants, moving beyond algebraic or norm-based classifications.

Findings

Identifies key invariants for topological equivalence classes

Distinguishes between compact and non-compact operators

Provides foundational steps towards a comprehensive classification

Abstract

In topological equivalence, a bounded linear operator between Banach spaces - we focus on the case of Hilbert spaces - is viewed as only acting linearly and continuously between them qua different spaces with the structure of linear topological space. For instance, invertible operators in Banach spaces (that is, isomorphisms among them) will make up one equivalence class for each class of isomorphic spaces. On the other hand, compact and non-compact operators, or operators with or without a kernel, clearly will not. We make some crucial steps towards describing invariants that will characterize these topological equivalence classes.