Generalized Hilbert Operator Acting on Bloch Type Spaces

2022/07/22 by Ye, Shanli, Zhou, Zhihui · 1 citation
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2207.11170

Abstract

Let μ be a positive Borel measure on the interval [0,1). For α>0, the Hankel matrix H_μ,α=(μ_n,k,α)_{n,k≥ 0} with entries μ_n,k,α=∫_[0,1)(Γ(n+α))/(n!Γ(α))t^n+kdμ(t) formally induces the operator H_μ,α(f)(z)=∑_n=0^∞(∑_k=0^∞ μ_{n, k,α} a_k)zⁿ on the space of all analytic functions f(z)=∑_k=0^∞a_kz^k in the unit disc \mathbbD. In this paper, we characterize the measures μ for which H_μ,α (α≥ 2) is a bounded (resp., compact) operator from the Bloch type space \mathscrB_β (0<β<∞) into \mathscrB_α-1. We also give a necessary condition for which H_μ,α is a bounded operator by acting on Bloch type spaces for general cases.

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