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Generalized limits and a mean ergodic theorem

100%

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.

Boundedness and growth orders of means of discrete and continuous semigroups of operators

81%

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

Studia Mathematica

2008

tom 187

nr 1

1-35

We discuss implication relations for boundedness and growth orders of Cesàro means and Abel means of discrete semigroups and continuous semigroups of linear operators. Counterexamples are constructed to show that implication relations between two Cesàro means of different orders or between Cesàro means and Abel means are in general strict, except when the space has dimension one or two.

Ograniczanie wyników

2 Studia Mathematica

2 Li Y.-C.

2 Shaw S.-Y.

1 Sato R.

1 2008

1 1996

Wyniki wyszukiwania

Generalized limits and a mean ergodic theorem

Boundedness and growth orders of means of discrete and continuous semigroups of operators