Convergence of Iterates of Normal Operators in L2 Spaces

Let (S2, , , m ) be a measure space with m being an sigma-finite positive measure and let N be a normal operator on L 2 (S2, , , m ). In this note, we study strong and almost everywhere convergences of the sequences {phi(N)nf ( N ) n f } n is an element of N in L 2 (S2, , , m ) spaces, where phi is a continuous function on the spectrum of N.