THE CESÀRO-LIKE OPERATOR ON SOME ANALYTIC FUNCTION SPACES

Let μ be a finite positive Borel measure on the interval [0,1) and f(z)= ∑ ⁡n=0∞anzn∈H(𝔻). The Cesàro-like operator is defined by Cμ(f)(z)=∑n=0∞(μn∑k=0nak)zn,z∈D, where, for n≥0, μn denotes the n-th moment of the measure μ, that is, μn= ∫ ⁡[0,1)tndμ(t). Let X and Y be subspaces of H(𝔻), the purpose of this paper is to study the action of 𝒞μ on distinct pairs (X,Y). The spaces considered in this paper are Hardy space Hp(0<p≤∞), Morrey space L2,λ(0<λ≤1), mean Lipschitz space, Bloch type space, etc.