Ergodic theorems of weak mixing type
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- by Lee K. Jones and Michael Lin PDF
- Proc. Amer. Math. Soc. 57 (1976), 50-52 Request permission
Abstract:
Given a linear contraction $T$ on a Banach space $X$ and $x \in X$, the convergence \[ \forall {x^ \ast } \in {X^ \ast }{N^{ - 1}}\sum \limits _{i = 1}^N {|\langle {x^ \ast },{T^i}x\rangle |} \xrightarrow [{N \to \infty }]{}0\] is shown to be equivalent to the convergence \[ \sup \limits _{||{x^ \ast }|| \leqslant 1} {N^{ - 1}}\sum \limits _{j = 1}^N {|\langle {x^ \ast },{T^{{k_j}}}x\rangle | \to 0} \] for every subsequence with ${k_j}/j$ bounded. A sufficient condition is that, for some $\{ {n_i}\} ,{T^{{n_i}}}x \to 0$ weakly.References
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Additional Information
- © Copyright 1976 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 57 (1976), 50-52
- MSC: Primary 47A35
- DOI: https://doi.org/10.1090/S0002-9939-1976-0405134-X
- MathSciNet review: 0405134