Maps preserving $m$- isometries on Hilbert space
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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 $\mathcal{H}$.
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 $\mathcal{H}$.
참고문헌 (9)
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