Finite-Branch Multipliers on a Hölder Algebra: Commutants, Norm Closures, and Essential Spectra
Abstract
Noninjective multiplication symbols allow commuting operators to mix function values along their fibers, while Hölder regularity constrains this mixing at branch points. We study these operators on the Banach algebra Aβ, 0<β<1, defined by a restricted interval-variation condition. For u(t)=t(1−t), we determine the full commutant {Mu}'=𝒞β, including operators whose reflection coefficients need not belong to Aβ. Separately, we characterize the operator-norm closure Jβ of the regular multiplication--reflection subalgebra of 𝒞β, obtain two-sided norm-distance estimates, and find a complemented copy of ℓ∞/c0 in the Banach-space quotient 𝒞β/Jβ. Every T∈𝒞β satisfies σ(T)=σap(T)=σusf(T)=σess(T), and Fredholmness is equivalent to invertibility. For multipliers with finitely many full bi-Lipschitz branches, we also characterize their full commutants and establish the same spectral equalities for every commuting operator. For the regular finite-branch algebra, we establish an exact norm-closure criterion using weighted local Hölder defects and separate row-difference conditions, even when several folds share a level.
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BibTeX
@misc{duru2026finitebranch,
title = {{Finite-Branch Multipliers on a Hölder Algebra: Commutants, Norm Closures, and Essential Spectra}},
author = {Hülya Duru and Serkan İlter and Arkady Kitover},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36217},
eprint = {2610.36217},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36217},
}
RIS
TY - UNPB
TI - Finite-Branch Multipliers on a Hölder Algebra: Commutants, Norm Closures, and Essential Spectra
AU - Hülya Duru
AU - Serkan İlter
AU - Arkady Kitover
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.36217
UR - https://arxiv.org/abs/2610.36217
AB - Noninjective multiplication symbols allow commuting operators to mix function values along their fibers, while Hölder regularity constrains this mixing at branch points. We study these operators on the Banach algebra A_β, 0<β<1, defined by a restricted interval-variation condition. For u(t)=t(1−t), we determine the full commutant {M_u}'=𝒞_β, including operators whose reflection coefficients need not belong to A_β. Separately, we characterize the operator-norm closure J_β of the regular multiplication--reflection subalgebra of 𝒞_β, obtain two-sided norm-distance estimates, and find a complemented copy of ℓ^∞/c_0 in the Banach-space quotient 𝒞_β/J_β. Every T∈𝒞_β satisfies σ(T)=σ_ap(T)=σ_usf(T)=σ_ess(T), and Fredholmness is equivalent to invertibility. For multipliers with finitely many full bi-Lipschitz branches, we also characterize their full commutants and establish the same spectral equalities for every commuting operator. For the regular finite-branch algebra, we establish an exact norm-closure criterion using weighted local Hölder defects and separate row-difference conditions, even when several folds share a level.
ER -