2022/01/24 by Mikael Lindström, Lindström, Mikael, Santeri Miihkinen +3 · 2 citations
Mathematics · #30H20 (Secondary) #47B38 (Primary) #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2201.09591
openalex publication_date 2022/01/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We compute the exact value of the essential norm of a generalized Hilbert matrix operator acting on weighted Bergman spaces Apv and weighted Banach spaces H^∞v of analytic functions, where v is a general radial weight. In particular, we obtain the exact value of the essential norm of the classical Hilbert matrix operator on standard weighted Bergman spaces Apα for p>2+α, α≥ 0, and on Korenblum spaces H^∞α for 0 < α< 1. We also cover the Hardy space Hp, 1 < p < ∞, case. In the weighted Bergman space case, the essential norm of the Hilbert matrix is equal to the conjectured value of its operator norm and similarly in the Hardy space case the essential norm and the operator norm coincide. We also compute the exact value of the norm of the Hilbert matrix on H^∞wα with weights wα(z)=(1-|z|)α for all 0 < α< 1. Also in this case, the values of the norm and essential norm coincide.