Document Type : Research Paper

Author

Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, P. O. Box 47416-1468, Babolsar, Iran.

Abstract

Let $\mathcal{A}$ and $\mathcal{B}$ be standard operator algebras on Banach spaces $\mathcal{X}$ and $\mathcal{Y}$, respectively. Let $\phi: \mathcal{A} \rightarrow \mathcal{B}$ be a bijective map. In this paper, we show that $\phi$ is completely preserving quadratic operator in both directions if and only if $\phi$ is 2-quadratic preserving operator in both directions and if and only if $\phi$ is either an isomorphism or (in the complex case) a conjugate isomorphism.

Keywords

###### ##### References
[1] M. Bresar and P. Semrl, On locally linearly dependent operators and derivation, Trans. Am. Math. Soc., 351 (1999), pp. 1257-1275.
[2] D. Hadwin and D. Larson, Completely rank nonincreasing linear maps, Ann. Funct. Anal., 199 (2003), pp. 263-277.
[3] D. Hadwin, J. Hou and H. Yousefi, Completely rank-nonincreasing linear maps on spaces of operators, Linear Algebra Appl., 383 (2004), pp. 213-232.
[4] R. Hosseinzadeh, I. Sharifi and A. Taghavi, Maps completely preserving fixed points and maps completely preserving kernel of operators, Anal. Math., 44 (2018), pp. 451-459.
[5] J. Hou and L. Huang, Characterizations of isomorphisms: maps completely preserving invertibility or spectrum, J. Appl. Math. Anal. Appl., 359 (2009), pp. 81-87.
[6] J. Hou and L. Huang, Maps completely preserving idempotents and maps completely preserving square-zero operators, Isr. J. Math., 176 (2010), pp. 363-380.
[7] L. Huang, Y. Liu, Maps completely preserving commutativity and maps completely preserving jordan-zero products, Isr. J. Math., 462 (2014), pp. 233-249.
[8] C.-K. Li, N.-K. Tsing, Linear preserver problems: a brief introduction and some special techniques, Linear Algebra Appl., 162-164 (1992), pp. 217-235.
[9] P.G. Ovchinnikov, Automorphisms of the poset of skew projections, J. Funct. Anal., 115 (1993), pp. 184-189.