Operators commuting with L∞]{Operators commuting with L∞ on Köthe--Bochner spaces
Abstract
Let E(X) be a Köthe--Bochner space of functions on a σ-finite measure space with values in a Banach space X. We study bounded operators on E(X) that commute with the operators of multiplication by functions from L∞. If X is separable, every such operator acts pointwise by an essentially bounded L(X)-valued function. For a nonseparable X, a lifting of L∞ provides a representation by L(X,X⋆⋆)-valued functions, and by L(X)-valued functions if X is complemented in X⋆⋆, with the sharp norm bound if X is a dual space. As an application, we show that if the transformation is aperiodic, then the spectral projections of invertible weighted composition operators with operator-valued weights are operators of multiplication by L(X)-valued functions whenever X is separable or complemented in X⋆⋆, and that, if moreover the measure is ergodic and dim X = m < ∞, the spectrum of such an operator has at most m connected components. Finally, for Lp(μ;X), 1 ≤ p < ∞, an aperiodic nonsingular transformation and a weight that is measurable in the operator norm, we prove that the spectral projections are operators of multiplication by projection-valued functions for an arbitrary Banach space X, for instance for X = c0(Γ) with Γ uncountable. The proof uses an inverse-closedness result for the algebra generated by norm-measurable multipliers and the composition operator.
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BibTeX
@misc{duru2026operators,
title = {{Operators commuting with $L^\infty$]Operators commuting with $L^\infty$ on Köthe--Bochner spaces}},
author = {Hulya Duru and Serkan Ilter and Arkady Kitover and Mehmet Orhon},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24853},
eprint = {2610.24853},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24853},
note = {CC0},
}
RIS
TY - UNPB
TI - Operators commuting with L^∞]{Operators commuting with L^∞ on Köthe--Bochner spaces
AU - Hulya Duru
AU - Serkan Ilter
AU - Arkady Kitover
AU - Mehmet Orhon
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24853
UR - https://arxiv.org/abs/2610.24853
AB - Let E(X) be a Köthe--Bochner space of functions on a σ-finite measure space with values in a Banach space X. We study bounded operators on E(X) that commute with the operators of multiplication by functions from L^∞. If X is separable, every such operator acts pointwise by an essentially bounded L(X)-valued function. For a nonseparable X, a lifting of L^∞ provides a representation by L(X,X^⋆⋆)-valued functions, and by L(X)-valued functions if X is complemented in X^⋆⋆, with the sharp norm bound if X is a dual space. As an application, we show that if the transformation is aperiodic, then the spectral projections of invertible weighted composition operators with operator-valued weights are operators of multiplication by L(X)-valued functions whenever X is separable or complemented in X^⋆⋆, and that, if moreover the measure is ergodic and dim X = m < ∞, the spectrum of such an operator has at most m connected components. Finally, for L^p(μ;X), 1 ≤ p < ∞, an aperiodic nonsingular transformation and a weight that is measurable in the operator norm, we prove that the spectral projections are operators of multiplication by projection-valued functions for an arbitrary Banach space X, for instance for X = c_0(Γ) with Γ uncountable. The proof uses an inverse-closedness result for the algebra generated by norm-measurable multipliers and the composition operator.
ER -