Two characterizations of automorphisms on B(X)

Peter Šemrl

Studia Mathematica (1993)

  • Volume: 105, Issue: 2, page 143-149
  • ISSN: 0039-3223

Abstract

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Let X be an infinite-dimensional Banach space, and let ϕ be a surjective linear map on B(X) with ϕ(I) = I. If ϕ preserves injective operators in both directions then ϕ is an automorphism of the algebra B(X). If X is a Hilbert space, then ϕ is an automorphism of B(X) if and only if it preserves surjective operators in both directions.

How to cite

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Šemrl, Peter. "Two characterizations of automorphisms on B(X)." Studia Mathematica 105.2 (1993): 143-149. <http://eudml.org/doc/215990>.

@article{Šemrl1993,
abstract = {Let X be an infinite-dimensional Banach space, and let ϕ be a surjective linear map on B(X) with ϕ(I) = I. If ϕ preserves injective operators in both directions then ϕ is an automorphism of the algebra B(X). If X is a Hilbert space, then ϕ is an automorphism of B(X) if and only if it preserves surjective operators in both directions.},
author = {Šemrl, Peter},
journal = {Studia Mathematica},
keywords = {automorphisms on ; automorphism; surjective operators},
language = {eng},
number = {2},
pages = {143-149},
title = {Two characterizations of automorphisms on B(X)},
url = {http://eudml.org/doc/215990},
volume = {105},
year = {1993},
}

TY - JOUR
AU - Šemrl, Peter
TI - Two characterizations of automorphisms on B(X)
JO - Studia Mathematica
PY - 1993
VL - 105
IS - 2
SP - 143
EP - 149
AB - Let X be an infinite-dimensional Banach space, and let ϕ be a surjective linear map on B(X) with ϕ(I) = I. If ϕ preserves injective operators in both directions then ϕ is an automorphism of the algebra B(X). If X is a Hilbert space, then ϕ is an automorphism of B(X) if and only if it preserves surjective operators in both directions.
LA - eng
KW - automorphisms on ; automorphism; surjective operators
UR - http://eudml.org/doc/215990
ER -

References

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  2. [2] M. D. Choi, A. A. Jafarian, and H. Radjavi, Linear maps preserving commutativity, Linear Algebra Appl. 87 (1987), 227-241. Zbl0615.15004
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  4. [4] J. C. Hou, Rank-preserving linear maps on B(X), Science in China (Ser. A) 32 (1989), 929-940. Zbl0686.47030
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  8. [8] J. Lindenstrauss, On nonseparable reflexive Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 967-970. Zbl0156.36403
  9. [9] M. Marcus and B. N. Moyls, Linear transformations on algebras of matrices, Canad. J. Math. 11 (1959), 61-66. Zbl0086.01703
  10. [10] M. Omladič, On operators preserving commutativity, J. Funct. Anal. 66 (1986), 105-122. Zbl0587.47051
  11. [12] R. I. Ovsepian and A. Pełczyński, Existence of a fundamental total and bounded biorthogonal sequence, Studia Math. 54 (1975), 149-159. Zbl0317.46019
  12. [13] C. Pearcy and D. Topping, Sums of small number of idempotents, Michigan Math. J. 14 (1967), 453-465. 
  13. [14] H. Radjavi and P. Rosenthal, Invariant Subspaces, Ergeb. Math. Grenzgeb. 77, Springer, Berlin 1973. 

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