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On permutability and submultiplicativity of spectral radius. (English) Zbl 0844.47003

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.

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
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