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# When does ‖f(A)‖ = f(‖A‖) hold true?

Publikationstyp

Journal Article

Publikationsdatum

2022

Sprache

English

Institut

Enthalten in

Volume

70

Issue

21

Start Page

7250

End Page

7263

Citation

Linear and Multilinear Algebra 21 (70): 7250-7263 (2022)

Publisher DOI

Scopus ID

Let a strongly stable norm (Formula presented.) on the set (Formula presented.) of complex n-by-n matrices be given, which means that (Formula presented.) for all (Formula presented.) and all (Formula presented.). Furthermore, let (Formula presented.) be a power series with nonnegative coefficients (Formula presented.) and radius of convergence R>0. If (Formula presented.), we additionally suppose that (Formula presented.). We aim to characterize those A with (Formula presented.), which fulfil (Formula presented.). We first show how to reduce the discussion of f to Neumann series. For matrix norms induced by uniformly convex vector norms, like the (Formula presented.) -norms, (Formula presented.), it follows from known results of the Daugavet equation (Formula presented.) that (Formula presented.) holds true if, and only if, (Formula presented.) is an eigenvalue of A, provided that (Formula presented.) for some (Formula presented.). Under adapted assumptions on the (Formula presented.) we prove that this equivalence remains true for the (Formula presented.) - and the (Formula presented.) -norm, for unitarily invariant matrix norms and for the numerical radius. We conjecture this equivalence to be valid for all strongly stable norms if (Formula presented.) for all (Formula presented.).