Rhaly operators between Dirichlet-type spaces: a complete classification
Abstract
Let η=(ηn)n≥0 be a complex sequence such that Fη(z)=∑n≥0ηnzn∈Hol(D), and let ℛ(η) be the associated Rhaly operator. We give a complete characterization of boundedness and compactness of ℛ(η):𝒟pα→𝒟qβ for 1<p,q<∞ and α,β>−1. The characterization is formulated in terms of the Hq-norms of the dyadic blocks of Fη. For −1<α<p−2, boundedness and compactness coincide and reduce to Fη∈𝒟qβ. For α=p−2 and for α>p−2, boundedness is characterized by membership in analytic truncated Besov spaces and analytic Besov spaces, respectively; compactness has the same characterization for q<p and is characterized by the corresponding little spaces for p≤ q. We also obtain operator norm and essential norm estimates. As applications, we obtain boundedness and compactness characterizations for Rhaly operators between weighted Bergman spaces and for Cesàro-type operators induced by positive measures. In particular, we characterize Cμ:Bp→ Bp, answering a question left open by Sun, Ye and Zhou and later noted by Tang. For each p>2, we also construct a symbol Fη∈ H∞∩λp1/p for which ℛ(η) is not bounded on Hp.
Related
- Rhaly and generalized Hilbert operators between spaces of analytic functions
- Generalized Stević–Sharma operators on weighted Dirichlet spaces
- Weighted Composition-differentiation operators on weighted Dirichlet spaces
- Some Properties of Products of Linear Fractional Composition Operators on Weighted Dirichlet Spaces
- Closed Range and Essential Norms of Composition Operators on Weighted Dirichlet Spaces
BibTeX
@misc{shi2026rhaly,
title = {{Rhaly operators between Dirichlet-type spaces: a complete classification}},
author = {Yecheng Shi},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15042},
eprint = {2609.15042},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15042},
}
RIS
TY - UNPB
TI - Rhaly operators between Dirichlet-type spaces: a complete classification
AU - Yecheng Shi
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.15042
UR - https://arxiv.org/abs/2609.15042
AB - Let η=(η_n)_n≥0 be a complex sequence such that F_η(z)=∑_n≥0η_nz^n∈Hol(D), and let ℛ_(η) be the associated Rhaly operator. We give a complete characterization of boundedness and compactness of ℛ_(η):𝒟^p_α→𝒟^q_β for 1<p,q<∞ and α,β>−1. The characterization is formulated in terms of the H^q-norms of the dyadic blocks of F_η. For −1<α<p−2, boundedness and compactness coincide and reduce to F_η∈𝒟^q_β. For α=p−2 and for α>p−2, boundedness is characterized by membership in analytic truncated Besov spaces and analytic Besov spaces, respectively; compactness has the same characterization for q<p and is characterized by the corresponding little spaces for p≤ q. We also obtain operator norm and essential norm estimates. As applications, we obtain boundedness and compactness characterizations for Rhaly operators between weighted Bergman spaces and for Cesàro-type operators induced by positive measures. In particular, we characterize C_μ:B^p→ B^p, answering a question left open by Sun, Ye and Zhou and later noted by Tang. For each p>2, we also construct a symbol F_η∈ H^∞∩λ^p_1/p for which ℛ_(η) is not bounded on H^p.
ER -