# Dirichlet series and uniform ergodic theorems for linear operators in Banach spaces

Studia Mathematica (2000)

• Volume: 141, Issue: 1, page 69-83
• ISSN: 0039-3223

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## Abstract

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We study the convergence properties of Dirichlet series for a bounded linear operator T in a Banach space X. For an increasing sequence $\mu ={\mu }_{n}$ of positive numbers and a sequence $f={f}_{n}$ of functions analytic in neighborhoods of the spectrum σ(T), the Dirichlet series for ${f}_{n}\left(T\right)$ is defined by D[f,μ;z](T) = ∑n=0∞ e-μnz fn(T), z∈ ℂ. Moreover, we introduce a family of summation methods called Dirichlet methods and study the ergodic properties of Dirichlet averages for T in the uniform operator topology.

## How to cite

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Yoshimoto, Takeshi. "Dirichlet series and uniform ergodic theorems for linear operators in Banach spaces." Studia Mathematica 141.1 (2000): 69-83. <http://eudml.org/doc/216774>.

@article{Yoshimoto2000,
abstract = {We study the convergence properties of Dirichlet series for a bounded linear operator T in a Banach space X. For an increasing sequence $μ = \{μ_n\}$ of positive numbers and a sequence $f = \{f_n\}$ of functions analytic in neighborhoods of the spectrum σ(T), the Dirichlet series for $\{f_n(T)\}$ is defined by D[f,μ;z](T) = ∑n=0∞ e-μnz fn(T), z∈ ℂ. Moreover, we introduce a family of summation methods called Dirichlet methods and study the ergodic properties of Dirichlet averages for T in the uniform operator topology.},
author = {Yoshimoto, Takeshi},
journal = {Studia Mathematica},
keywords = {Abel method; uniform ergodicity; nilpotent operator; Dirichlet averages; power-boundedness},
language = {eng},
number = {1},
pages = {69-83},
title = {Dirichlet series and uniform ergodic theorems for linear operators in Banach spaces},
url = {http://eudml.org/doc/216774},
volume = {141},
year = {2000},
}

TY - JOUR
AU - Yoshimoto, Takeshi
TI - Dirichlet series and uniform ergodic theorems for linear operators in Banach spaces
JO - Studia Mathematica
PY - 2000
VL - 141
IS - 1
SP - 69
EP - 83
AB - We study the convergence properties of Dirichlet series for a bounded linear operator T in a Banach space X. For an increasing sequence $μ = {μ_n}$ of positive numbers and a sequence $f = {f_n}$ of functions analytic in neighborhoods of the spectrum σ(T), the Dirichlet series for ${f_n(T)}$ is defined by D[f,μ;z](T) = ∑n=0∞ e-μnz fn(T), z∈ ℂ. Moreover, we introduce a family of summation methods called Dirichlet methods and study the ergodic properties of Dirichlet averages for T in the uniform operator topology.
LA - eng
KW - Abel method; uniform ergodicity; nilpotent operator; Dirichlet averages; power-boundedness
UR - http://eudml.org/doc/216774
ER -

## References

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2. [2] N. Dunford, Spectral theory I. Convergence to projection, Trans. Amer. Math. Soc. 54 (1943), 185-217. Zbl0063.01185
3. [3] N. Dunford and J. T. Schwartz, Linear Operators I: General Theory, Pure Appl. Math., Interscience, New York, 1958. Zbl0084.10402
4. [4] E. Hille, Remarks on ergodic theorems, Trans. Amer. Math. Soc. 57 (1945), 246-269. Zbl0063.02017
5. [5] K. B. Laursen and M. Mbekhta, Operators with finite chain length and ergodic theorem, Proc. Amer. Math. Soc. 123 (1995), 3443-3448. Zbl0849.47008
6. [6] M. Lin, On the uniform ergodic theorem, Proc. Amer. Math. Soc. 43 (1974), 337-340. Zbl0252.47004
7. [7] M. Lin, On the uniform ergodic theorem II, ibid. 46 (1974), 217-225.
8. [8] M. Mbekhta et J. Zemánek, Sur le théorème ergodique uniforme et le spectre, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), 1155-1158.
9. [9] G. Szegő, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. 23, Providence, 1939.
10. [10] A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, 2nd ed., Wiley, 1980. Zbl0501.46003
11. [11] T. Yoshimoto, Uniform and strong ergodic theorems in Banach spaces, Illinois J. Math. 42 (1998), 525-543; Correction, ibid. 43 (1999), 800-801. Zbl0924.47005

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