Mean Lipschitz spaces and a generalized Hilbert operator

2017/07/27 by Noel Merchán
Computer Science · Mathematics · #Advanced Banach Space Theory #Borel measure #Generalization #Hilbert matrix #Hilbert space #Holomorphic and Operator Theory #Lipschitz continuity #Measure (data warehouse) #Operator (biology) #Optimization and Variational Analysis #Order (exchange) #Unit interval #math.CV #msc:30H10

paper · pdf · doi:10.1007/s13348-018-0217-y

published as Collect. Math. (2019) 70 (1), 59-69 · 11 pages, 0 figures

arxiv created 2017/07/27 · openalex created_date 2017/08/08 · openalex publication_date 2018/02/28 · arxiv updated 2019/01/28 · openalex updated_date 2026/08/05

Abstract

If μ is a positive Borel measure on the interval [0, 1) we let \mathcal H_μ be the Hankel matrix \mathcal H_μ=(μ_{n, k})_{n,k≥ 0} with entries μ_{n, k}=μ_n+k, where, for n = 0, 1, 2, … , μ_n denotes the moment of order n of μ. This matrix induces formally the operator H_μ(f)(z)= ∑_n=0^∞(∑_k=0^∞ μ_n,ka_k)zⁿ on the space of all analytic functions f(z)=∑_k=0^∞ a_kz^k, in the unit disc \mathbbD . This is a natural generalization of the classical Hilbert operator. In this paper we study the action of the operators \mathcal H_μ on mean Lipschitz spaces of analytic functions.

Citations

• A Hankel Matrix Acting on Spaces of Analytic Functions