Applications of Operator Semigroups To Fourier

Dedicated to the memory of Al Cliiord, a wonderful mathematician and a wonderful man. Abstract Of concern are semigroups of linear norm one operators on Hilbert space of the form (discrete case) T = fT n : n = 0; 1; 2; g or (continuous case) T = fT (t) : t 0g: Using ergodic theory and Hilbert-Schmidt operators, the Cess aro limits (as n! 1) of j< T n f; f >j 2 ; j< T (n)f; f >j 2 are computed (with n 2 Z + or n 2 R +). Specializing the Hilbert space to be L 2 (T;) (discrete case) or L 2 (R;) (continuous case) where is a Borel probability measure on the circle group or the line, the Cess aro limit of j ^ (n) j 2 (as n! 1 ; with n 2 Z or n 2 R) is obtained and interpreted. Extensions to T M and R M are given. Finally, we discuss recent operator theoretic extensions from a Hilbert to a Banach space context.