The iterates of a contraction and its adjoint

Abstract

We prove that when T is a contraction on Hilbert space the size of | ( ( T ∗ ) n h , g ) | |({({T^\ast })^n}h,g)| is controlled by that of lim sup | ( T n h , g ) | \lim \sup |({T^n}h,g)| . We give an application to Fourier-Stieltjes coefficients. Important in the proof is a generalization of the technique of orthogonal projection.