On the Bergman space norm of the Cesàro operator (Q1924990)

We determine the exact values of the norm and the spectrum for the Cesàro operator \(C_p\) on the Bergman spaces \(A^p\) for \(p\geq 4\) and give estimates for the norm and spectrum for \(1\leq p<4\). For \(p\geq 4\) the values are \(|C_p|=p/2\) and \(\sigma(C_p)= \{z:|z-p/4|\leq p/4\}\). We also find, as a byproduct, the spectra of certain composition operators on \(A^p\). Further a general necessary condition is given for \(C\) to be bounded on certain Banach spaces of analytic functions.