Maps on matrices that preserve the spectral radius distance

Rajendra Bhatia; Peter Šemrl; A. Sourour

Studia Mathematica (1999)

  • Volume: 134, Issue: 2, page 99-110
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Bhatia, Rajendra, Šemrl, Peter, and Sourour, A.. "Maps on matrices that preserve the spectral radius distance." Studia Mathematica 134.2 (1999): 99-110. <http://eudml.org/doc/216632>.

@article{Bhatia1999,
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.},
author = {Bhatia, Rajendra, Šemrl, Peter, Sourour, A.},
journal = {Studia Mathematica},
keywords = {maps on matrices; spectral radius},
language = {eng},
number = {2},
pages = {99-110},
title = {Maps on matrices that preserve the spectral radius distance},
url = {http://eudml.org/doc/216632},
volume = {134},
year = {1999},
}

TY - JOUR
AU - Bhatia, Rajendra
AU - Šemrl, Peter
AU - Sourour, A.
TI - Maps on matrices that preserve the spectral radius distance
JO - Studia Mathematica
PY - 1999
VL - 134
IS - 2
SP - 99
EP - 110
AB - 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.
LA - eng
KW - maps on matrices; spectral radius
UR - http://eudml.org/doc/216632
ER -

References

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  1. [1] R. Bhatia and P. Šemrl, Approximate isometries on Euclidean spaces, Amer. Math. Monthly 104 (1997), 497-504. Zbl0901.46016
  2. [2] A. A. Jafarian and A. R. Sourour, Spectrum-preserving linear maps, J. Funct. Anal. 66 (1986), 255-261. Zbl0589.47003
  3. [3] M. Jerison, The space of bounded maps into a Banach space, Ann. of Math. 52 (1950), 307-327. Zbl0038.27301
  4. [4] S. Mazur et S. Ulam, Sur les transformations isométriques d'espaces vectoriels normés, C. R. Acad. Sci. Paris 194 (1932), 946-948. Zbl58.0423.01
  5. [5] P. Šemrl, Linear maps that preserve the nilpotent operators, Acta Sci. Math. (Szeged) 61 (1995), 523-534. Zbl0843.47024

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