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Generalized limits and a mean ergodic theorem

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Li Y.-C., Shaw S.-Y.

Studia Mathematica

1996

tom 121

nr 3

207-219

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.

Ograniczanie wyników

1 Studia Mathematica

1 Li Y.-C.

1 Shaw S.-Y.

1 1996

Wyniki wyszukiwania

Generalized limits and a mean ergodic theorem