Rhaly Operators Acting on Hardy, Bergman, and Dirichlet Spaces

Abstract In this article we address the question of characterizing the sequences of complex numbers $$(\eta )=\{ \eta _n\}_{n=0}^\infty $$ ( η ) = { η n } n = 0 ∞ whose associated Rhaly operator $${\mathcal {R}}_{(\eta )}$$ R ( η ) is bounded or compact on the Hardy spaces $$H^p$$ H p ( $$1\le p - 1 ). We give a number of conditions which are either necessary or sufficient for the boundedness (compactness) of $${\mathcal {R}}_{(\eta )}$$ R ( η ) on these spaces. These conditions have to do with the membership in certain mean Lipschitz spaces of analytic functions of the function $$F_{(\eta )}$$ F ( η ) defined by $$F_{(\eta )}(z)=\sum _{n=0}^\infty \eta _nz^n$$ F ( η ) ( z ) = ∑ n = 0 ∞ η n z n