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We compute the exact value of the essential norm of a generalized Hilbertmatrix operator acting on weighted Bergman spaces $A^p_v$ and weighted Banachspaces $H^\infty_v$ of analytic functions, where $v$ is a general radialweight. In particular, we obtain the exact value of the essential norm of theclassical 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 normof the Hilbert matrix is equal to the conjectured value of its operator normand similarly in the Hardy space case the essential norm and the operator normcoincide. 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 normcoincide.

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