Let T be a power-bounded operator on a (real or complex) Banach space. We study the convergence of the one-sided ergodic Hubert transform limn, ∑k=1n Tkx/k. We prove that weak and strong convergence are equivalent, and in a reflexive space also supn ||∑k=1n Tkx/k|| < ∞ is equivalent to the convergence. We also show that - ∑k=1∞ Tk/k (which converges on (I - T)X) is precisely the infinitesimal generator of the semigroup (J - T)τ /|(I - T)X.
- Ergodic hubert transform
- Measure preserving transformations
- Operator power series
- Semigroup of fractional powers
ASJC Scopus subject areas
- Mathematics (all)