A remark on the trace of some Riesz operators
type: Item
instance of: scholarly article
title: A remark on the trace of some Riesz operators
zbMATH Open document ID: 0817.47023
DOI: 10.1007/BF01202069
author: Yeneng Sun
published in: Archiv der Mathematik
publication date: 1995-04-03T00:00:00Z
review text: An asymptotic estimate for the trace of iterations of a certain class of Riesz operators is provided. In particular, we obtain in the following theorem a new spectral radius formula for such operators. Theorem: Let \(T\) be a bounded linear operator on a Banach space \(X\) such that \(T^ h\) is in the class \(S^ a_ 1\) of operators for some positive integer \(h\). Let \(\lambda_ n\), \(n\in \mathbb{N}\) be the sequence of eigenvalues of \(T\) ordered in the following way: the eigenvalues are non- increasing in absolute value, i.e. \(| \lambda_ n |\geq | \lambda_{n+1} |\), and each eigenvalue is repeated as often as its multiplicity indicates. Let \(r\) be the spectral radius of \(T\). Assume that \(r>0\). Let \(k\) be the number of eigenvalues with modulus \(r\). Then \[ \varlimsup_{m\to \infty} {{| \text{tr } T^ m|} \over {r^ m}} =k. \] As a corollary, we improve a result concerning the trace of matrices which is originated from the study of symbolic dynamics. Let \(A\) be a \(k\times k\) matrix all of whose entries are 0 or 1. Then \(A\) induces a shift transformation \(\sigma_ A\) on a closed subspace \(X_ A\) of the product space \(\{1, \dots, k\}^ \mathbb{Z}\). The pair \((X_ A, \sigma_ A)\) is called a sub-shift of finite type which has a long history in dynamical systems. Assume that \(\lambda_ 1\) is an eigenvalue of \(A\) with the maximal modulus. Then \(\varlimsup_{n\to \infty} {1\over n} \log| \text{tr } A^ n|= \log| \lambda_ 1 |\), which is the entropy of the shift mapping \(\sigma_ A\). Let \(m\) be the number of eigenvalues with the same modulus as \(\lambda_ 1\). Then the above theorem yields that \(\varlimsup_{n\to \infty} [{1\over n}\log | \text{tr } A^ n|- \log | \lambda_ 1 |]= \log m\), which provides a better estimate than the known result \(\varlimsup_{n\to \infty} [{1\over n}\log | \text{tr } A^ n|- \log| \lambda_ 1 |]=0\).
Mathematics Subject Classification ID: 47B06
Mathematics Subject Classification ID: 47A35
zbMATH DE Number: 734494
zbMATH Keywords: entropy
zbMATH Keywords: eigenvalue with the maximal modulus
zbMATH Keywords: Riesz operators
zbMATH Keywords: spectral radius formula
zbMATH Keywords: trace of matrices
zbMATH Keywords: symbolic dynamics
zbMATH Keywords: sub-shift of finite type
MaRDI profile type: Publication
cites work: Eigenvalue distribution of compact operators
cites work: Q4767350
cites work: Q3289300
cites work: Zur Fredholmschen Theorie in lokalkonvexen Räumen
cites work: Q3744918
cites work: Q5332803
full work available at URL: https://doi.org/10.1007/bf01202069
OpenAlex ID: W1968276892
description: scientific article; zbMATH DE number 734494