# Generalized limits and a mean ergodic theorem

Studia Mathematica (1996)

- Volume: 121, Issue: 3, page 207-219
- ISSN: 0039-3223

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topLi, Yuan-Chuan, and Shaw, Sen-Yen. "Generalized limits and a mean ergodic theorem." Studia Mathematica 121.3 (1996): 207-219. <http://eudml.org/doc/216352>.

@article{Li1996,

abstract = {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.},

author = {Li, Yuan-Chuan, Shaw, Sen-Yen},

journal = {Studia Mathematica},

keywords = {Banach limits; $L$-limits; states; numerical radius; reflexive space; mean ergodic theorem; -limit; -limit; abstract mean ergodic theorem},

language = {eng},

number = {3},

pages = {207-219},

title = {Generalized limits and a mean ergodic theorem},

url = {http://eudml.org/doc/216352},

volume = {121},

year = {1996},

}

TY - JOUR

AU - Li, Yuan-Chuan

AU - Shaw, Sen-Yen

TI - Generalized limits and a mean ergodic theorem

JO - Studia Mathematica

PY - 1996

VL - 121

IS - 3

SP - 207

EP - 219

AB - 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.

LA - eng

KW - Banach limits; $L$-limits; states; numerical radius; reflexive space; mean ergodic theorem; -limit; -limit; abstract mean ergodic theorem

UR - http://eudml.org/doc/216352

ER -

## References

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