# Generalized limits and a mean ergodic theorem

Studia Mathematica (1996)

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

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## Abstract

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

## How to cite

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Li, 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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1. [1] Z. U. Ahmad and Mursaleen, An application of Banach limits, Proc. Amer. Math. Soc. (1) 103 (1988), 244-246.
2. [2] F. F. Bonsall and J. Duncan, Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras, London Math. Soc. Lecture Note Ser. 2, Cambridge Univ. Press, 1971. Zbl0207.44802
3. [3] F. F. Bonsall and J. Duncan, Numerical Ranges II, London Math. Soc. Lecture Ser. 10, Cambridge Univ. Press, 1973. Zbl0262.47001
4. [4] A. Brunel, H. Fong, and L. Sucheston, An ergodic superproperty of Banach spaces defined by a class of matrices, Proc. Amer. Math. Soc. 49 (1975), 373-378. Zbl0272.47008
5. [5] D. van Dulst, Reflexive and Superreflexive Banach Spaces, MCT, 1982.
6. [6] U. Krengel, Ergodic Theorems, de Gruyter, 1985.
7. [7] G. G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80 (1948), 167-190. Zbl0031.29501
8. [8] Mursaleen, On some new invariant matrix methods of summability, Quart. J. Math. Oxford (2) 34 (1983), 77-86.
9. [9] R. A. Raimi, Invariant means and invariant matrix methods of summability, Duke Math. J. 30 (1963), 81-94. Zbl0125.03201
10. [10] P. Schaefer, Infinite matrices and invariant means, Proc. Amer. Math. Soc. 36 (1972), 104-110. Zbl0255.40003
11. [11] P. Schaefer, Mappings of positive integers and subspaces of m, Portugal. Math. 38 (1979), 29-38. Zbl0503.47028
12. [12] S.-Y. Shaw, Mean ergodic theorems and linear functional equations, J. Funct. Anal. 87 (1989), 428-441. Zbl0704.47006
13. [13] A. M. Sinclair, The states of a Banach algebra generate the dual, Proc. Edinburgh Math. Soc. (2) 17 (1971), 193-200. Zbl0233.46062
14. [14] K. Yosida and S. Kakutani, Operator-theoretical treatment of Markoff's process and mean ergodic theorem, Ann. of Math. 42 (1941), 188-228. Zbl0024.32402

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