Generalized limits and a mean ergodic theorem

• Autorzy: Yuan-Chuan Li , Sen-Yen Shaw
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Banach limits   $L$-limits   states   numerical radius   reflexive space   mean ergodic theorem  

Kategorie tematyczne

• 47A35: Ergodic theory
• 47A12: Numerical range, numerical radius

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1996
• Tom: 121
• Numer: 3
• Strony: 207-219

Daty

wydano

1996

otrzymano

1995-02-02

poprawiono

1996-07-15

Twórcy

autor

Yuan-Chuan Li

• Department of Mathematics, Chung Yuan University, Chung-Li, Taiwan

autor

Sen-Yen Shaw

• Department of Mathematics, National Central University, Chung-Li, Taiwan ,

Bibliografia

• [1] Z. U. Ahmad and Mursaleen, An application of Banach limits, Proc. Amer. Math. Soc. (1) 103 (1988), 244-246.
• [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.
• [3] F. F. Bonsall and J. Duncan, Numerical Ranges II, London Math. Soc. Lecture Ser. 10, Cambridge Univ. Press, 1973.
• [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.
• [5] D. van Dulst, Reflexive and Superreflexive Banach Spaces, MCT, 1982.
• [6] U. Krengel, Ergodic Theorems, de Gruyter, 1985.
• [7] G. G. Lorentz, A contribution to the theory of divergent sequences, Acta Math. 80 (1948), 167-190.
• [8] Mursaleen, On some new invariant matrix methods of summability, Quart. J. Math. Oxford (2) 34 (1983), 77-86.
• [9] R. A. Raimi, Invariant means and invariant matrix methods of summability, Duke Math. J. 30 (1963), 81-94.
• [10] P. Schaefer, Infinite matrices and invariant means, Proc. Amer. Math. Soc. 36 (1972), 104-110.
• [11] P. Schaefer, Mappings of positive integers and subspaces of m, Portugal. Math. 38 (1979), 29-38.
• [12] S.-Y. Shaw, Mean ergodic theorems and linear functional equations, J. Funct. Anal. 87 (1989), 428-441.
• [13] A. M. Sinclair, The states of a Banach algebra generate the dual, Proc. Edinburgh Math. Soc. (2) 17 (1971), 193-200.
• [14] K. Yosida and S. Kakutani, Operator-theoretical treatment of Markoff's process and mean ergodic theorem, Ann. of Math. 42 (1941), 188-228.