In this paper, we calculate the exact value of the norm of the Hilbert matrix operator \({\mathcal {H}}\) from the logarithmically weighted Korenblum space \(H^\infty _{\alpha ,\log }\) into Korenblum space \(H^\infty _\alpha \) , and from the Hardy space \(H^\infty \) to the classical Bloch space \({\mathcal {B}}\) . Furthermore, we compute the precise value of the norm on the logarithmically weighted Korenblum space \(H^\infty _{\alpha ,\log }\) , and obtain both the lower and upper bounds of the norm on \(\alpha \) -Bloch space \({\mathcal {B}}^{\alpha }\) . Finally, in the context of mapping from the Korenblum space \(H^\infty _\alpha \) to the \((\alpha +1)\) -Bloch space \({\mathcal {B}}^{\alpha +1}\) , we establish the norm of \({\mathcal {H}}\) .
