Ces\xe0ro-like operators acting on spaces of analytic functions

Abstract

Let {mathbb {D}} be the unit disc in {mathbb {C}}. If mu is a finite positive Borel measure on the interval [0, 1) and f is an analytic function in {mathbb {D}}, f(z)=sum _{n=0}^infty a_nz^n (zin {mathbb {D}}), we define Cμ(f)(z)=∑n=0∞μn∑k=0nakzn,z∈D,\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} {\\mathcal {C}}_\\mu (f)(z)= \\sum _{n=0}^\\infty \\mu _n\\left( \\sum _{k=0}^na_k\\right) z^n,\\quad z\\in {\\mathbb {D}}, \\end{aligned}$$\\end{document}where, for nge 0, mu _n denotes the n-th moment of the measure mu , that is, mu _n=int _{[0, 1)}t^ndmu (t). In this way, {mathcal {C}}_mu becomes a linear operator defined on the space {mathrm{Hol}}({mathbb {D}}) of all analytic functions in {mathbb {D}}. We study the action of the operators {mathcal {C}}_mu on distinct spaces of analytic functions in {mathbb {D}}, such as the Hardy spaces H^p, the weighted Bergman spaces A^p_alpha , BMOA, and the Bloch space {mathcal {B}}.