Exact essential norm of generalized Hilbert matrix operators on classical analytic function spaces

TL;DR

This paper precisely calculates the essential norm of generalized Hilbert matrix operators on various classical analytic function spaces, confirming conjectured values and revealing cases where the essential and operator norms are equal.

Contribution

It provides exact formulas for the essential norm of Hilbert matrix operators on weighted Bergman, Hardy, and Korenblum spaces, extending previous bounds and conjectures.

Findings

Essential norm equals the operator norm in many cases.

Exact values computed for classical and weighted spaces.

Confirmed conjectured norms for Hilbert matrix operators.

Abstract

We compute the exact value of the essential norm of a generalized Hilbert matrix operator acting on weighted Bergman spaces $A^p_v$ and weighted Banach spaces $H^\infty_v$ 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 $A^p_\alpha$ for $p>2+\alpha, \, \alpha \ge 0,$ and on Korenblum spaces $H^\infty_\alpha$ for $0 < \alpha < 1.$ We also cover the Hardy space $H^p, \, 1 < p < \infty,$ 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 $H^\infty_{w_\alpha}$ with weights $w_\alpha(z)=(1-|z|)^\alpha$ for all $0 < \alpha < 1$. Also in this case, the values of the norm and essential norm coincide.