| Title:
|
Nörlund means of the sequence of the iterates of a bounded linear operator, and spectral properties (English) |
| Author:
|
Burlando, Laura |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
150 |
| Issue:
|
3 |
| Year:
|
2025 |
| Pages:
|
415-443 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
We are concerned here with relating the spectral properties of a bounded linear operator $T$ on a Banach space to the behaviour of the means $(1/{s(n)})\sum _{k=0}^n(\Delta s)(n-k)T^k$, where $s$ is a nondecreasing sequence of positive real numbers, and $\Delta $ denotes the inverse of the automorphism on the vector space of scalar sequences which maps each sequence into the sequence of its partial sums. In a previous paper, we obtained a uniform ergodic theorem for the means above, under the hypotheses $\lim _{n\rightarrow \infty }s(n)=\infty$, $\lim _{n\rightarrow \infty }{s(n+1)}/{s(n)}=1$, and $\Delta ^qs\in \ell _1$ for a positive integer $q$: indeed, we proved that if $T^n/s(n)$ converges to zero in the uniform operator topology for such a sequence $s$, then the averages above converge in the same topology if and only if 1 is either in the resolvent set of $T$, or a simple pole of the resolvent function of $T$. In this paper, we prove that if $\liminf _{n\rightarrow \infty }{s(n+1)}/{s(n)}=1$, and the averages above converge in the uniform operator topology, then 1 is either in the resolvent set of $T$, or a simple pole of the resolvent function of $T$. The converse is not true, even if the sequence $s$ satisfies all the hypotheses of the theorem recalled above, except membership of $\Delta ^qs$ in $\ell _1$ for a positive integer $q$. We also prove that if $\lim _{n\rightarrow \infty }\root n\of {s(n)}=1$, and the function $h_s(z)=\sum _{n=0}^{\infty }s(n)z^n$ has no zeros in the open unit disk, then operator norm boundedness of the averages of the sequence $T^n$induced by $s$ implies that the spectral radius of $T$ is less than or equal to $1$. This result fails if the assumption about $h_s$ is dropped. Indeed, it may happen that the averages converge in the uniform operator topology for a sequence $s$ satisfying $\lim _ {n\rightarrow \infty }s(n)=\infty $, $\lim _{n\rightarrow \infty } {s(n+1)}/{s(n)}=1$, and $\Delta ^qs\in l_1$ for a positive integer $q$, and nevertheless the spectral radius of $T$ is strictly larger than 1. (English) |
| Keyword:
|
bounded linear operator |
| Keyword:
|
uniform ergodic theorem |
| Keyword:
|
Nörlund means of operator iterates |
| Keyword:
|
spectrum |
| Keyword:
|
pole of the resolvent |
| MSC:
|
47A10 |
| MSC:
|
47A35 |
| DOI:
|
10.21136/MB.2024.0067-24 |
| . |
| Date available:
|
2025-09-26T14:35:37Z |
| Last updated:
|
2025-09-26 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/153085 |
| . |
| Reference:
|
[1] Allan, G. R., Ransford, T. J.: Power-dominated elements in a Banach algebra.Stud. Math. 94 (1989), 63-79. Zbl 0705.46021, MR 1008239, 10.4064/sm-94-1-63-79 |
| Reference:
|
[2] Burlando, L.: A uniform ergodic theorem for some Nörlund means.Commun. Math. Anal. 21 (2018), 1-34. Zbl 07002173, MR 3866091 |
| Reference:
|
[3] Conway, J. B.: Functions of One Complex Variable.Graduate Texts in Mathematics 11. Springer, New York (1978). Zbl 0277.30001, MR 0503901, 10.1007/978-1-4612-6313-5 |
| Reference:
|
[4] Dunford, N.: Spectral theory. I. Convergence to projections.Trans. Am. Math. Soc. 54 (1943), 185-217. Zbl 0063.01185, MR 0008642, 10.1090/S0002-9947-1943-0008642-1 |
| Reference:
|
[5] Dunford, N.: Spectral theory.Bull. Am. Math. Soc. 49 (1943), 637-651. Zbl 0063.01184, MR 0008643, 10.1090/S0002-9904-1943-07965-7 |
| Reference:
|
[6] Ed-dari, E.: On the $(C,\alpha)$ uniform ergodic theorem.Stud. Math. 156 (2003), 3-13. Zbl 1052.47005, MR 1961058, 10.4064/sm156-1-1 |
| Reference:
|
[7] Halmos, P. R.: A Hilbert Space Problem Book.D. Van Nostrand, Princeton (1967). Zbl 0144.38704, MR 0208368 |
| Reference:
|
[8] Hille, E.: Remarks on ergodic theorems.Trans. Am. Math. Soc. 57 (1945), 246-269. Zbl 0063.02017, MR 0012212, 10.1090/S0002-9947-1945-0012212-0 |
| Reference:
|
[9] Hille, E., Phillips, R. S.: Functional Analysis and Semi-Groups.American Mathematical Society Colloquium Publications 31. AMS, Providence (1957). Zbl 0078.10004, MR 0089373, 10.1090/coll/031 |
| Reference:
|
[10] Laursen, K. B., Mbekhta, M.: Operators with finite chain length and the ergodic theorem.Proc. Am. Math. Soc. 123 (1995), 3443-3448. Zbl 0849.47008, MR 1277123, 10.1090/S0002-9939-1995-1277123-9 |
| Reference:
|
[11] Lin, M.: On the uniform ergodic theorem.Proc. Am. Math. Soc. 43 (1974), 337-340. Zbl 0252.47004, MR 0417821, 10.1090/S0002-9939-1974-0417821-6 |
| Reference:
|
[12] Lin, M., Shoikhet, D., Suciu, L.: Remarks on uniform ergodic theorems.Acta Sci. Math. 81 (2015), 251-283. Zbl 1363.47015, MR 3381884, 10.14232/actasm-012-307-4 |
| Reference:
|
[13] Mbekhta, M., Zemánek, J.: Sur le théorème ergodique uniforme et le spectre.C. R. Acad. Sci. Paris, Sér. I 317 (1993), 1155-1158 French. Zbl 0792.47006, MR 1257230 |
| Reference:
|
[14] Rudin, W.: Principles of Mathematical Analysis.McGraw-Hill, New York (1976). Zbl 0346.26002, MR 0385023 |
| Reference:
|
[15] Taylor, A. E., Lay, D. C.: Introduction to Functional Analysis.John Wiley & Sons, New York (1980). Zbl 0501.46003, MR 0564653 |
| Reference:
|
[16] Yoshimoto, T.: Uniform and strong ergodic theorems in Banach spaces.Ill. J. Math. 42 (1998), 525-543. Zbl 0924.47005, MR 1648580, 10.1215/ijm/1255985459 |
| Reference:
|
[17] Zygmund, A.: Trigonometric Series Volumes I & II Combined.Cambridge Mathematical Library. Cambridge University Press, Cambridge (2002). Zbl 1084.42003, MR 1963498, 10.1017/CBO9781316036587 |
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