For a given linear operator L on with ∥L∥ = 1 and L(1) = 1, a notion of limit, called the L-limit, is defined for bounded sequences in a normed linear space X. In the case where L is the left shift operator on and , the definition of L-limit reduces to Lorentz’s definition of σ-limit, which is described by means of Banach limits on . We discuss some properties of L-limits, characterize reflexive spaces in terms of existence of L-limits of bounded sequences, and formulate a version of the abstract mean ergodic theorem in terms of L-limits. A theorem of Sinclair on the form of linear functionals on a unital normed algebra in terms of states is also generalized.
@article{bwmeta1.element.bwnjournal-article-smv121i3p207bwm,
author = {Yuan-Chuan Li and Sen-Yen Shaw},
title = {Generalized limits and a mean ergodic theorem},
journal = {Studia Mathematica},
volume = {119},
year = {1996},
pages = {207-219},
zbl = {0869.47008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv121i3p207bwm}
}
Li, Yuan-Chuan; Shaw, Sen-Yen. Generalized limits and a mean ergodic theorem. Studia Mathematica, Tome 119 (1996) pp. 207-219. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv121i3p207bwm/
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