The one-sided ergodic Hilbert transform in Banach spaces

摘要

Let T be a power-bounded operator on a (real or complex) Banach space.We study the convergence of the one-sided ergodic Hilbert transform lim_n(k=1)~n(T~kx~x)/kWe prove that weak and strong convergence are equivalent, and in a reflexive space also sup_n|| _(k=1)~nTkx||< is equivalent to the convergence. We also show that - Tkk (which converges on (I - T)X) is precisely The infinitesimal generator of the semigroup(I - T)~r |(I-T)X.