Maps on matrices that preserve the spectral radius distance

• Autorzy: Rajendra Bhatia , Peter Šemrl , A. R. Sourour
• Języki publikacji: EN

Abstrakty

EN

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.

Kategorie tematyczne

• 15A18: Eigenvalues, singular values, and eigenvectors
• 47A10: Spectrum, resolvent

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1999
• Tom: 134
• Numer: 2
• Strony: 99-110

Daty

wydano

1999

otrzymano

1997-06-27

Twórcy

autor

Rajendra Bhatia

• Indian Statistical Institute, New Delhi 110 016, India

autor

Peter Šemrl

• Department of Mathematics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia

autor

A. R. Sourour

• Department of Mathematics and Statistics, University of Victoria, Victoria, B.C., Canada V8W 3P4

Bibliografia

• [1] R. Bhatia and P. Šemrl, Approximate isometries on Euclidean spaces, Amer. Math. Monthly 104 (1997), 497-504.
• [2] A. A. Jafarian and A. R. Sourour, Spectrum-preserving linear maps, J. Funct. Anal. 66 (1986), 255-261.
• [3] M. Jerison, The space of bounded maps into a Banach space, Ann. of Math. 52 (1950), 307-327.
• [4] S. Mazur et S. Ulam, Sur les transformations isométriques d'espaces vectoriels normés, C. R. Acad. Sci. Paris 194 (1932), 946-948.
• [5] P. Šemrl, Linear maps that preserve the nilpotent operators, Acta Sci. Math. (Szeged) 61 (1995), 523-534.