A remark on the trace of some Riesz operators
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 \(\varlimsup_{n\to \infty} [{1\over n}\log | \text{tr } A^ n|- \log| \lambda_ 1 |]=0\).
scientific article; zbMATH DE number 734494