Abstract
Let X be an infinite–-dimensional complex Banach space and denote by ℬ(X) the algebra of all bounded linear operators acting on X. It is shown that a surjective additive map Φ from ℬ(X) onto itself preserves similarity in both directions if and only if there exist a scalar c, a bounded invertible linear or conjugate linear operator A and a similarity invariant additive functional φ on ℬ(X) such that either Φ(T) = cATA −1 + φ(T)I for all T, or Φ(T) = cAT*A −1 + φ(T)I for all T. In the case where X has infinite multiplicity, in particular, when X is an infinite–dimensional Hilbert space, the above similarity invariant additive functional φ is always zero.
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References
Bai, Z. F., Hou, J. C.: Distance preserving maps on C*-algebras, preprint
Bai, Z. F., Hou, J. C., Xu, Z. B.: Maps preserving numerical radius distance on C*-algebras. Studia Math., 162, 97–104 (2004)
Bai, Z. F., Hou, J. C.: Numerical radius distance preserving maps on ℬ(H). Proc. Amer. Math. Soc., 132, 1453–1461 (2004)
Cui, J. L., Hou, J. C.: Non-linear numerical radius isometries on atomic nest algebras and diagonal algebras. J. Func. Anal., 206, 414–448 (2004)
Petek, T., Semrl, P.: Adjacency preserving maps on matrices and operators. Proc. Roy. Soc. Edinb., 132A, 661–684 (2002)
Hiai, F.: Similarity preserving linear maps on matrices. Linear Algebra Appl., 97, 127–139 (1987)
Lim, M. H.: A note on similarity preserving linear maps on matrices. Linear Algebra Appl., 190, 229–233 (1993)
Ji, G. X., Du, H. K.: Similarity-invariant subspaces and similarity-preserving linear maps. Acta Mathematica Sinica, English Series, 18(3), 489–498 (2002)
Ji, G. X.: Similarity-preserving linear maps on B(H). Linear Algebras Appl., 360, 249–257 (2003)
Ji, G. X.: Asymptotical similarity-preserving linear maps on B(H). Linear Algebra Appl., 368, 371–378 (2003)
Bai, Z. F., Hou, J. C.: Additive maps preserving nilpotent operators or spectral radius. Acta Mathematica Sinica, English Series, 21(5), 1167–1182 (2005)
Šemrl, P.: Linear maps that preserve the nilpotent operators. Acta. Sci. Math. (Szeged), 61, 1523–1534 (1995)
Wilansky, A.: Subalgebras of B(X). Proc. Amer. Math. Soc., 29, 355–360 (1971)
Hadwin, L. B.: Local multiplications on algebras spanned by idempotents. Linear and Multilinear Algebra, 37, 259–263 (1994)
Hou, J. C., Cui, J. L.: Introduction to the Linear Maps on Operator Algebras, Science Press, Beijing, 2002
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Hou, J.C., Zhang, X.L. Additive Maps Preserving Similarity of Operators on Banach Spaces. Acta Math Sinica 22, 179–186 (2006). https://doi.org/10.1007/s10114-005-0564-y
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DOI: https://doi.org/10.1007/s10114-005-0564-y