Composition Semigroups on the Besov Spaces
Abstract
We study semigroups of composition operators acting on the Besov spaces Bp, where they exhibit some new behaviors relative to many classical spaces. Often for a Banach space X of analytic functions on the unit disk, the maximal closed space of strong continuity, [ φt, X ], exists for every semigroup { φt } of analytic self-maps of the disk, and the question whether [φt , X ] equals X itself has an answer independent of {φt}. Such is the case for the Hardy and Bergman spaces, Bloch, BMOA, and H∞. For the disk algebra A, [φt , A ] = A precisely when {φt} ⊂ A. For Bp with p ≥ 2, every {φt} ⊂ Bp and always [ φt, Bp ] = Bp, but this fails when 1 < p < 2. We give an example where {φt} ⊂ Bp and yet the induced composition operators {Ct} are not bounded on Bp and we do not know if [φt,Bp] exists. If it does exist, it cannot be equal to Bp. Under the hypothesis that there is a uniform bound for the operator norms of the {Ct}, 0 ≤ t ≤ 1, we characterize the semigroups { φt } such that [ φt, Bp ] = Bp.
Versions
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BibTeX
@misc{anderson2026composition,
title = {{Composition Semigroups on the Besov Spaces}},
author = {Austin Anderson and Mirjana Jovovic and Wayne Smith},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.21878},
eprint = {2607.21878},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.21878},
}
RIS
TY - UNPB
TI - Composition Semigroups on the Besov Spaces
AU - Austin Anderson
AU - Mirjana Jovovic
AU - Wayne Smith
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2607.21878
UR - https://arxiv.org/abs/2607.21878
AB - We study semigroups of composition operators acting on the Besov spaces B_p, where they exhibit some new behaviors relative to many classical spaces. Often for a Banach space X of analytic functions on the unit disk, the maximal closed space of strong continuity, [ φ_t, X ], exists for every semigroup { φ_t } of analytic self-maps of the disk, and the question whether [φ_t , X ] equals X itself has an answer independent of {φ_t}. Such is the case for the Hardy and Bergman spaces, Bloch, BMOA, and H^∞. For the disk algebra A, [φ_t , A ] = A precisely when {φ_t} ⊂ A. For B_p with p ≥ 2, every {φ_t} ⊂ B_p and always [ φ_t, B_p ] = B_p, but this fails when 1 < p < 2. We give an example where {φ_t} ⊂ B_p and yet the induced composition operators {C_t} are not bounded on B_p and we do not know if [φ_t,B_p] exists. If it does exist, it cannot be equal to B_p. Under the hypothesis that there is a uniform bound for the operator norms of the {C_t}, 0 ≤ t ≤ 1, we characterize the semigroups { φ_t } such that [ φ_t, B_p ] = B_p.
ER -