Maps preserving $m$- isometries on Hilbert space
Main Article Content
Abstract
Let $\mathcal{H}$ be a complex Hilbert space and $\mathcal{B}(\mathcal{H})$ the algebra of all bounded linear operators on $\mathcal{H}$. In this paper, we prove that if $\varphi:\mathcal{B}(\mathcal{H})\to \mathcal{B}(\mathcal{H})$ is a unital surjective bounded linear map, which preserves $m$- isometries $m=1, 2$ in both directions, then there are unitary operators $U, V\in \mathcal{B}(\mathcal{H})$ such that
\begin{eqnarray*}
\varphi(T)=UTV\quad {\rm or}\quad \varphi(T)=UT^{tr}V
\end{eqnarray*}
for all $T\in \mathcal{B}(\mathcal{H})$, where $T^{tr}$ is the transpose of $T$ with respect to an arbitrary but fixed orthonormal basis of $\mathcal{H}$.
Article Details
References
[1] F. Bayart, m-isometries on Banach spaces, Math. Nachr. 284 (17-18) (2011), 2141–2147. Google Scholar
[2] A. Chahbi and S. Kabbaj, Linear maps preserving G-unitary operators in Hilbert space, Arab J. Math. Sci. 21 (1) (2015), 109–117. Google Scholar
[3] I. N. Herstein, Topics in ring theory, University of Chicago Press, Chicago, 1969. Google Scholar
[4] A. Majidi and M. Amyari, Maps preserving quasi- isometries on Hilbert C∗-modules, Rocky Mountain J. Math. 48 (4) (2018). Google Scholar
[5] A. Majidi and M. Amyari, On maps that Preserve ∗-product of operators in B(H), Tamsui Oxford J. Math. Sci. 33 (1) (2019). Google Scholar
[6] L. Molnar, Selected preserver problems on algebraic structures of linear operators and on function spaces, Springer, 1895. Google Scholar
[7] G. J. Murphy, C∗-algebras and operator theory, Academic Press Inc, London, 1990. Google Scholar
[8] M. Rais, The unitary group preserving maps (the in finite-dimensional case), Linear Multilinear Algebra, 20 (1987), 337–345. Google Scholar
[9] Y. N. Wei and G. X. Ji, Maps preserving partial isometries of operator pencils, (Chinese) Acta Math. Sci. Ser. A Chin. Ed. 36 (3) (2016), 413–424. Google Scholar