Geodesic
Generalized limits and a mean ergodic theorem
Studia Mathematica, Tome 121 (1996) no. 3, pp. 207-219

Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences

For a given linear operator L on $ℓ^∞$ 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 $ℓ^∞$ and $X = ℓ^∞$, the definition of L-limit reduces to Lorentz's definition of σ-limit, which is described by means of Banach limits on $ℓ^∞$. 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.
DOI : 10.4064/sm-121-3-207-219
Keywords: Banach limits, $L$-limits, states, numerical radius, reflexive space, mean ergodic theorem

Yuan-Chuan Li 1 ; Sen-Yen Shaw 1

1 
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Yuan-Chuan Li; Sen-Yen Shaw. Generalized limits and a mean ergodic theorem. Studia Mathematica, Tome 121 (1996) no. 3, pp. 207-219. doi: 10.4064/sm-121-3-207-219

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