Cesàro-like operators acting on spaces of analytic functions

Abstract

AbstractLet $${\mathbb {D}}$$ D be the unit disc in $${\mathbb {C}}$$ C . If $$\mu $$ μ is a finite positive Borel measure on the interval [0, 1) and f is an analytic function in $${\mathbb {D}}$$ D , $$f(z)=\sum _{n=0}^\infty a_nz^n$$ f ( z ) = ∑ n = 0 ∞ a n z n ($$z\in {\mathbb {D}}$$ z ∈ D ), we define $$\begin{aligned} {\mathcal {C}}_\mu (f)(z)= \sum _{n=0}^\infty \mu _n\left( \sum _{k=0}^na_k\right) z^n,\quad z\in {\mathbb {D}}, \end{aligned}$$ C μ ( f ) ( z ) = ∑ n = 0 ∞ μ n ∑ k = 0 n a k z n , z ∈ D , where, for $$n\ge 0$$ n ≥ 0 , $$\mu _n$$ μ n denotes the n-th moment of the measure $$\mu $$ μ , that is, $$\mu _n=\int _{[0, 1)}t^nd\mu (t).$$ μ n = ∫ [ 0 , 1 ) t n d μ ( t ) . In this way, $${\mathcal {C}}_\mu $$ C μ becomes a linear operator defined on the space $${\mathrm{Hol}}({\mathbb {D}})$$ Hol ( D ) of all analytic functions in $${\mathbb {D}}$$ D . We study the action of the operators $${\mathcal {C}}_\mu $$ C μ on distinct spaces of analytic functions in $${\mathbb {D}}$$ D , such as the Hardy spaces $$H^p$$ H p , the weighted Bergman spaces $$A^p_\alpha $$ A α p , BMOA, and the Bloch space $${\mathcal {B}}$$ B .