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.
@article{bwmeta1.element.bwnjournal-article-smv105i2p143bwm,
author = {Peter \v Semrl},
title = {Two characterizations of automorphisms on B(X)},
journal = {Studia Mathematica},
volume = {104},
year = {1993},
pages = {143-149},
zbl = {0810.47001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv105i2p143bwm}
}
Šemrl, Peter. Two characterizations of automorphisms on B(X). Studia Mathematica, Tome 104 (1993) pp. 143-149. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv105i2p143bwm/
[00000] [1] M. D. Choi, D. Hadwin, E. Nordgren, H. Radjavi, and P. Rosenthal, On positive linear maps preserving invertibility, J. Funct. Anal. 59 (1984), 462-469. | Zbl 0551.46040
[00001] [2] M. D. Choi, A. A. Jafarian, and H. Radjavi, Linear maps preserving commutativity, Linear Algebra Appl. 87 (1987), 227-241. | Zbl 0615.15004
[00002] [3] C. Davis and P. Rosenthal, Solving linear operator equations, Canad. J. Math. 26 (1974), 1384-1389. | Zbl 0297.47011
[00003] [4] J. C. Hou, Rank-preserving linear maps on B(X), Science in China (Ser. A) 32 (1989), 929-940. | Zbl 0686.47030
[00004] [5] A. A. Jafarian and A. R. Sourour, Spectrum-preserving linear maps, J. Funct. Anal. 66 (1986), 255-261. | Zbl 0589.47003
[00005] [6] I. Kaplansky, Infinite Abelian Groups, University of Michigan Press, Ann Arbor 1954. | Zbl 0057.01901
[00006] [7] K. B. Laursen and P. Vrbová, Some remarks on the surjectivity spectrum of linear operators, Czechoslovak Math. J. 39 (1989), 730-739. | Zbl 0715.47003
[00007] [8] J. Lindenstrauss, On nonseparable reflexive Banach spaces, Bull. Amer. Math. Soc. 72 (1966), 967-970. | Zbl 0156.36403
[00008] [9] M. Marcus and B. N. Moyls, Linear transformations on algebras of matrices, Canad. J. Math. 11 (1959), 61-66. | Zbl 0086.01703
[00009] [10] M. Omladič, On operators preserving commutativity, J. Funct. Anal. 66 (1986), 105-122. | Zbl 0587.47051
[00010] [12] R. I. Ovsepian and A. Pełczyński, Existence of a fundamental total and bounded biorthogonal sequence, Studia Math. 54 (1975), 149-159. | Zbl 0317.46019
[00011] [13] C. Pearcy and D. Topping, Sums of small number of idempotents, Michigan Math. J. 14 (1967), 453-465.
[00012] [14] H. Radjavi and P. Rosenthal, Invariant Subspaces, Ergeb. Math. Grenzgeb. 77, Springer, Berlin 1973.