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=₀β_kT^kx when {β_k} is a Kaluza sequence with divergent sum such that β_k→0 and ∑^∞k=₀β_kz^k converges in the open unit disk. We prove that weak and strong convergence are equivalent, and in a reflexive space also sup_n∥∑ⁿ_k=0β_kT^kx∥<∞ 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=0β_nTⁿx, 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 {β_k} is also completely monotone (i.e. a Hausdorff moment sequence) and β_k=O(1/k), the norm convergence of ∑^∞_k=0β_kT^kƒimplies a.e. convergence.

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