On convergence of power series of Lp contractions

Let T be a power-bounded operator on a (real or complex) Banach space. We study the convergence of the power series ∑∞k=0βkTkx when {βk} is a Kaluza sequence with divergent sum such that βk→0 and ∑∞k=0βkzk converges in the open unit disk. We prove that weak and strong convergence are equivalent, and in a reflexive space also supn∥∑nk=0βkTkx∥<∞ is equivalent to the convergence of the series. The last assertion is proved also when T is a mean ergodic contraction of L1.

For normal operators on a Hilbert space we obtain a spectral characterization of the convergence of ∑∞n=0βnTnx, and a sufficient condition expressed in terms of norms of the ergodic averages, which in some cases is also necessary.

For T Dunford–Schwartz of a σ-finite measure space or a positive contraction of Lp, 1<p<∞, we prove that when {βk} is also completely monotone (i.e. a Hausdorff moment sequence) and βk=O(1/k), the norm convergence of ∑∞k=0βkTkf implies a.e. convergence.

For T a positive contraction of Lp, p>1, f∈Lp and β∈R, we show that if the series ∑∞n=0(log(n+1))β(n+1)1−1/rTnf converges in Lp-norm for some r∈(pp−1,∞], then it converges a.e.