A Derivative-Hilbert operator acting on Hardy spaces

2022/06/24 by Shanli Ye, Ye, Shanli, Guanghao Feng +1 · 1 citation
Mathematics · #Holomorphic and Operator Theory #Meromorphic and Entire Functions #Advanced Harmonic Analysis Research

paper · pdf · doi:10.48550/arxiv.2206.12024

Abstract

Let μ be a positive Borel measure on the interval [0,1). The Hankel matrix H_μ= (μ_n,k)_n,k≥0 with entries μ_n,k= μ_n+k, where μ_n=∫ _[0,1)tⁿdμ(t), induces formally the operator DH_μ(f)(z)=∑_n=0^∞ (∑_k=0^∞ μ_n,ka_k)(n+1)zⁿ on the space of all analytic function f(z)=∑_k=0^ ∞ a_k zⁿ in the unit disc \mathbbD. We characterize those positive Borel measures on [0,1) such that DH_μ(f)(z)= ∫_[0,1) (f(t))/((1-tz)²) dμ(t) for all in Hardy spaces H^p(0 p and q≥ 1). We also study the analogous problem in Hardy spaces H^p(1≤ p≤ 2).

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