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Fall 1999 On modulated ergodic theorems for Dunford-Schwartz operators
Michael Lin, James Olsen, Arkady Tempelman
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Illinois J. Math. 43(3): 542-567 (Fall 1999). DOI: 10.1215/ijm/1255985110


We investigate sequences of complex numbers $\mathbf{a} = \{a_{k}\}$ for which the modulated averages $\frac{1}{n}\sum^{n}_{k=1}{a_{k}T^{k} f}$ converge in norm for every weakly almost periodic linear operator $T$ in a Banach space. For Dunford-Schwartz operators on probability spaces, we study also the a.e. convergence in $L_{p}$. The limit is identified in some special cases, in particular when $T$ is a contraction in a Hilbert space, or when $\mathbf{a} = \{S^{k}\phi(\xi)\}$ for some positive Dunford-Schwartz operator $S$ on a Lebesgue space and $\phi \in L_{2}$. We also obtain necessary and sufficient conditions on $\mathbf{a}$ for the norm convergence of the modulated averages for every mean ergodic power bounded $T$, and identify the limit.


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Michael Lin. James Olsen. Arkady Tempelman. "On modulated ergodic theorems for Dunford-Schwartz operators." Illinois J. Math. 43 (3) 542 - 567, Fall 1999.


Published: Fall 1999
First available in Project Euclid: 19 October 2009

zbMATH: 0939.47008
MathSciNet: MR1700609
Digital Object Identifier: 10.1215/ijm/1255985110

Primary: 47A35
Secondary: 28D05

Rights: Copyright © 1999 University of Illinois at Urbana-Champaign


Vol.43 • No. 3 • Fall 1999
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