On Nörlund summation and Ergodic Theory, with applications to power series of Hilbert contractions

2017/07/01 by Cuny, Christophe, Weber, Michel
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1707.00104

Abstract

We show that if \bf a=(a_n)_{n∈ \N} is a good weight for the dominated weighted ergodic theorem in L^p, p>1, then the Nörlund matrix N_{\bf a}=\a_i-j/A_i\_{0≤ j≤ i}, A_i=∑_k=0ⁱ |a_k| is bounded on ℓ^p(\N). We study the regularity (convergence in norm, almost everywhere) of operators in ergodic theory: power series of Hilbert contractions, and power series ∑_{n∈ \N} a_nP_nf of L²-contractions, and establish similar tight relations with the Nörlund operator associated to the modulus coefficient sequence (|a_n|)_{n∈ \N}.

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