Applications of operator semigroups to Fourier analysis

Of concern are semigroups of linear norm one operators on Hilbert space of the form (discrete case) T={T-n\n=0,1,2,...} or (continuous case) T={T(t)\t greater than or equal to 0}. Using ergodic theory and Hilbert-Schmidt operators, the Cesaro limits (as n-->infinity) of \[T(n)f,f]\(2), \[T(n)f,f]\(2) are computed (with n is an element of Z(+) or n is an element of R(+)). Specializing the Hilbert space to be L(2)(T,mu) (discrete case) or L(2)(R,mu) (continuous case) where mu is a Borel probability measure on the circle group or the line, the Cesaro limit of \<(mu)over cap>(n)\(2) (as n-->+/-infinity, with n is an element of Z or n is an element of 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.