RT Journal Article
T1 Rhaly Operators Acting on Hardy, Bergman, and Dirichlet Spaces
A1 Galanopoulos, Petros
A1 Girela-Álvarez, Daniel
K1 Hardy, Espacios de
K1 Dirichlet, Series de
K1 Número complejos
AB In this article we address the question of characterizing the sequences of complexnumbers (η) = {ηn}∞n=0 whose associated Rhaly operator R(η) is bounded or compacton the Hardy spaces H p (1 ≤ p < ∞), on the Bergman spaces Apα, and on theDirichlet spaces Dpα (1 ≤ p < ∞, α > −1). We give a number of conditions whichare either necessary or sufficient for the boundedness (compactness) of R(η) on thesespaces. These conditions have to do with the membership in certain mean Lipschitzspaces of analytic functions of the function F(η) defined by F(η)(z) = ∞n=0 ηn zn(z ∈ D). We prove that if 2 ≤ p < ∞ and ηn = O  1n , then R(η) is bounded on H p.However, there exists a sequence (η) with ηn = O  1n such that the operator R(η) isnot bounded on H p for 1 ≤ p < 2. We deal also with the derivative-Hardy spaces.For p > 0 the derivative-Hardy space S p consists of those functions f , analytic inthe unit disc D, such that f   ∈ H p. We prove that if 1 ≤ p < ∞ and 1 < q < ∞then R(η) is a bounded operator from S p into Sq if and only if it is compact and thishappens if and only if F(η) ∈ Sq .
PB Springer
YR 2026
FD 2026-02
LK https://hdl.handle.net/10630/45733
UL https://hdl.handle.net/10630/45733
LA eng
NO Galanopoulos, P., Girela, D. Rhaly Operators Acting on Hardy, Bergman, and Dirichlet Spaces. J Geom Anal 36, 115 (2026). https://doi.org/10.1007/s12220-026-02361-9
NO Funding for open access charge: Universidad de Málaga / CBUA
DS RIUMA. Repositorio Institucional de la Universidad de Málaga
RD 17 mar 2026