EUDML

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Generalized limits and a mean ergodic theorem

Yuan-Chuan Li; Sen-Yen Shaw — 1996

Studia Mathematica

For a given linear operator L on $ℓ^{\infty}$ 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 $ℓ^{\infty}$ and $X = ℓ^{\infty}$ , the definition of L-limit reduces to Lorentz’s definition of σ-limit, which is described by means of Banach limits on $ℓ^{\infty}$ . 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...

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

Yuan-Chuan Li; Ryotaro Sato; Sen-Yen Shaw — 2008

Studia Mathematica

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.