Weak and Weak* Operators in Non-Separable Banach Spaces: Topological Properties, Convergence and Structural Insights

Abstract

Juan Alberto Molina Garcia*

This article develops a rigorous framework for the study of weak and weak* operators in nonseparable Banach spaces (NSBS), where many of the foundational results of classical functional analysis fail or require significant reformulation. While the separable case is governed by compactness principles such as the Banach–Alaoglu theorem, the Eberlein– Šmulian theorem, and Rosenthal’s characterization of weakly compact sets, these rely heavily on metrizability and sequential compactness, both of which are absent in non-separable settings. Consequently, the extension of weak and weak* operator theory to NSBS is not straightforward and requires a careful topological re-evaluation.

The main objective of this work is to provide generalized notions of weak compactness, weak*continuity, and convergence of operators in NSBS, formulated in terms of nets rather than sequences, and to establish stability results for these properties under perturbations. We show that weak compactness can still be characterized locally on separable subspaces, but its global behavior diverges substantially from classical expectations. In particular, we prove that the failure of sequential compactness leads to an abundance of counterexamples in spaces such as l∞, L∞ ([0,1]) yC(βN), where operators exhibit residual behaviors not captured by traditional weak compactness criteria.

New results are presented regarding the structure of weakly compact and weak*-continuous operators in NSBS. These include a reformulated Dunford–Pettis property, characterisations of weakly compact operators in terms of invariant separable subspaces, and stability theorems under compact and norm perturbations. We also analyse the fragility of weak operator convergence in NSBS, showing that weak operator topology (WOT) perturbations may fail to preserve spectral or topological features, but that partial resolvent stability can still be obtained.

The article contributes to functional analysis in three principal ways: (i) by clarifying the limitations of classical theorems in the non-separable setting; (ii) by introducing generalised tools, based on local analysis and net convergence, that preserve part of the operator-theoretic structure; and (iii) by identifying new avenues of research at the intersection of weak compactness, operator algebras, and duality theory. Finally, we outline potential interdisciplinary applications.

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