ArticleOriginal scientific text
Title
Maps on matrices that preserve the spectral radius distance
Authors 1, 2, 3
Affiliations
- Indian Statistical Institute, New Delhi 110 016, India
- Department of Mathematics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
- Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., Canada V8W 3P4
Abstract
Let ϕ be a surjective map on the space of n×n complex matrices such that r(ϕ(A)-ϕ(B))=r(A-B) for all A,B, where r(X) is the spectral radius of X. We show that ϕ must be a composition of five types of maps: translation, multiplication by a scalar of modulus one, complex conjugation, taking transpose and (simultaneous) similarity. In particular, ϕ is real linear up to a translation.
Bibliography
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- A. A. Jafarian and A. R. Sourour, Spectrum-preserving linear maps, J. Funct. Anal. 66 (1986), 255-261.
- M. Jerison, The space of bounded maps into a Banach space, Ann. of Math. 52 (1950), 307-327.
- S. Mazur et S. Ulam, Sur les transformations isométriques d'espaces vectoriels normés, C. R. Acad. Sci. Paris 194 (1932), 946-948.
- P. Šemrl, Linear maps that preserve the nilpotent operators, Acta Sci. Math. (Szeged) 61 (1995), 523-534.
Pages:
99-110
Main language of publication
English
Received
1997-06-27
Published
1999
Exact and natural sciences