Spectral conditions and band reducibility of operators
Let \(M\) be a (complex) matrix with non-negative entries. Then the diagonal of \(M\) exhibits the eigenvalues of \(M\) with their multiplicity if and only if there exists a permutation matrix \(P\) such that \(P^{-1}MP\) is triangular with the same diagonal as \(M\). This pleasant fact is not very complicated to check (see, for instance, Theorem \(5.1.7\) in [\textit{H. Radjavi} and \textit{P. Rosenthal}, Simultaneous triangularization. Universitext. New York, NY: Springer (2000; Zbl 0981.15007)]). The authors of this remarkable paper try to extend this result to positive operators on \(L^{p}\)-spaces. They obtain some affirmative results under extra conditions of compactness, viz., the operator under consideration (or, at least some power of it) has to be compact. In this regard, examples are given to justify such additional hypotheses. A typical result is that, if \(T\) is a band-irreducible bounded positive operator \(T\) on \(L^{p}\left( X,\mu\right) \), then the spectral radius of any (proper) compact band compression of \(T\) is strictly smaller than that of \(T\). The reader is encouraged to read the whole paper to discover alternative (and very interesting) results.
scientific article; zbMATH DE number 6073816