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

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.