Inner multipliers, similarity and invariant subspaces for a class of Harmonically weighted Dirichlet spaces
Abstract
We consider the harmonically weighted Dirichlet spaces D(μ) induced by Borel measures which are mutually absolutely continuous with respect to the Lebesgue measure m on the unit circle 𝕋. Let H2 and D denote the Hardy space and the Dirichlet space on the unit disc 𝔻, respectively. For any f∈ H2, let mf denote the measure defined by dmf(ζ)=|f(ζ)|2 dm(ζ). Our first result provides a characterization of functions in D(mf), when f satisfies the condition |f'(z)|2=O(1/((1−|z|)1−ε)) for some ε>0. We then study the multiplier algebras of these spaces with particular emphasis on inner multipliers. Specifically, when f' is bounded, we obtain an explicit description of all singular inner functions in the multiplier algebra of D(mf) in terms of the radial zero set of f. As an application, we prove that if f',g' are bounded and the radial zero sets of f and g are different, then the operators (Mz, D(mf)) and (Mz, D(mg)) are not similar. This generalizes a result of Richter showing that the operators (Mz, D(mz−1)) and (Mz, D) are not similar. Then, we classify all invariant subspaces ℳ of (Mz, D(μ)) for which Mz|ℳ is similar to (Mz, D(μ)), where μ is any measure mutually absolutely continuous with respect to m. Finally, we study a special case of the problem of finding all functions g∈ D for which gD(mg) is a closed invariant subspace of (Mz, D). A key ingredient in many of our results is the Richter-Sundberg formula for the local Dirichlet integral.
Related
- Numerical range and invariant subspaces of weighted composition operators on the Hardy space of Dirichlet series
- The critical line for modulation invariant bilinear multipliers
- Noncommutative commutant lifting, interpolation, and shift-invariant subspaces
- Projections and minimal invariant subspaces in the Hardy space over the bidisk
- Invariant subspaces for free linearizations of Lipschitz maps
BibTeX
@misc{ghara2026inner,
title = {{Inner multipliers, similarity and invariant subspaces for a class of Harmonically weighted Dirichlet spaces}},
author = {Soumitra Ghara and Avipsa Patra},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.16988},
eprint = {2609.16988},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.16988},
}
RIS
TY - UNPB
TI - Inner multipliers, similarity and invariant subspaces for a class of Harmonically weighted Dirichlet spaces
AU - Soumitra Ghara
AU - Avipsa Patra
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.16988
UR - https://arxiv.org/abs/2609.16988
AB - We consider the harmonically weighted Dirichlet spaces D(μ) induced by Borel measures which are mutually absolutely continuous with respect to the Lebesgue measure m on the unit circle 𝕋. Let H^2 and D denote the Hardy space and the Dirichlet space on the unit disc 𝔻, respectively. For any f∈ H^2, let m_f denote the measure defined by dm_f(ζ)=|f(ζ)|^2 dm(ζ). Our first result provides a characterization of functions in D(m_f), when f satisfies the condition |f'(z)|^2=O(1/((1−|z|)^1−ε)) for some ε>0. We then study the multiplier algebras of these spaces with particular emphasis on inner multipliers. Specifically, when f' is bounded, we obtain an explicit description of all singular inner functions in the multiplier algebra of D(m_f) in terms of the radial zero set of f. As an application, we prove that if f',g' are bounded and the radial zero sets of f and g are different, then the operators (M_z, D(m_f)) and (M_z, D(m_g)) are not similar. This generalizes a result of Richter showing that the operators (M_z, D(m_z−1)) and (M_z, D) are not similar. Then, we classify all invariant subspaces ℳ of (M_z, D(μ)) for which M_z|_ℳ is similar to (M_z, D(μ)), where μ is any measure mutually absolutely continuous with respect to m. Finally, we study a special case of the problem of finding all functions g∈ D for which gD(m_g) is a closed invariant subspace of (M_z, D). A key ingredient in many of our results is the Richter-Sundberg formula for the local Dirichlet integral.
ER -