Multipliers on Noncommutative Orlicz Spaces

TL;DR

This paper develops a general theory of multiplication operators between noncommutative Orlicz spaces $L^{\varphi}(\widetilde{\mathcal{M}})$ over semifinite von Neumann algebras. It first establishes broad existence criteria based on a generalized Hausdorff–Young inequality, extending known finite-measure results to the noncommutative setting. It then analyzes common multiplier techniques, including rescaling, measure-equivalence effects, and the relation to composition operators via Jordan $*$-morphisms, highlighting key departures from the $L^p$ case. Finally, it characterizes compact multipliers, showing that on non-atomic algebras compactness essentially forces the multiplier to be zero, with nontrivial compact examples arising only from finite-type direct summands. Collectively, the results deepen our understanding of operator theory on noncommutative Orlicz spaces and clarify how multiplier theory diverges from the classical $L^p$ theory.

Abstract

We establish very general criteria for the existence of multiplication operators between noncommutative Orlicz spaces $L^{ψ_0}(\tM)$ and $L^{ψ_1}(\tM)$. We then show that these criteria contain existing results, before going on to briefly look at the extent to which the theory of multipliers on Orlicz spaces differs from that of $L^p$-spaces. In closing we describe the compactness properties of such operators.

Theorems & Definitions (24)

• Lemma 1.1
• proof
• Lemma 1.2: LM
• Proposition 1.3: LM
• Proposition 1.4
• Theorem 2.1: KR
• Theorem 2.2
• proof
• Corollary 2.3
• proof