Longstaff, W. E.; Radjavi, H. On permutability and submultiplicativity of spectral radius. (English) Zbl 0844.47003 Can. J. Math. 47, No. 5, 1007-1022 (1995). Summary: Let \(r(T)\) denote the spectral radius of the operator \(T\) acting on a complex Hilbert space \(H\). Let \({\mathcal S}\) be a multiplicative semigroup of operators on \(H\). We say that \(r\) is permutable on \({\mathcal S}\) if \(r(ABC)= r(BAC)\), for every \(A,B, C\in {\mathcal S}\). We say that \(r\) is submultiplicative on \({\mathcal S}\) if \(r(AB)\leq r(A) r(B)\), for every \(A,B\in {\mathcal S}\). It is known that, if \(r\) is permutable on \({\mathcal S}\), then it is submultiplicative. We show that the converse holds in each of the following cases: (i) \({\mathcal S}\) consists of compact operators, (ii) \({\mathcal S}\) consists of normal operators, (iii) \({\mathcal S}\) is generated by orthogonal projections. Cited in 6 Documents MSC: 47A10 Spectrum, resolvent 47A15 Invariant subspaces of linear operators 47D03 Groups and semigroups of linear operators 20M20 Semigroups of transformations, relations, partitions, etc. 15A30 Algebraic systems of matrices Keywords:spectral radius; permutable; submultiplicative; compact operators; normal operators; orthogonal projections PDFBibTeX XMLCite \textit{W. E. Longstaff} and \textit{H. Radjavi}, Can. J. Math. 47, No. 5, 1007--1022 (1995; Zbl 0844.47003) Full Text: DOI