Generalize Hilbert operator acting on Dirichlet spaces

2022/08/01 by Zhao, Liyun, Wang, Zhenyou, Su, Zhirong
#Complex Variables (math.CV) #FOS: Mathematics

paper · doi:10.48550/arxiv.2208.00951

Abstract

Let μ be a positive Borel measure on the interval [0,1). For γ>0, the Hankel matrix H_μ,γ=(μ_n,k)_n,k≥0 with entries μ_n,k=μ_n+k, where μ_n+k=∫₀^∞t^n+kdμ(t). formally induces the operator H_μ,γ=∑_n=0^∞(∑_k=0^∞μ_n,ka_k)(Γ(n+γ))/(n!Γ(γ))zⁿ, on the space of all analytic functions f(z)=∑_k=0^∞a_kz^k in the unit disc \mathbbD. Following ideas from \citeauthor3 and \citeauthor4, in this paper, for 0≤α<2, 2≤β<4, γ≥1. we characterize the measure μ for which H_μ,γ is bounded(resp.,compact)from D_α into D_β.

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