Maps preserving $m$- isometries on Hilbert space

Alireza Majidi

doi:10.11568/kjm.2019.27.3.735

Korean J. Math. Vol. 27 No. 3 (2019) pp.735-741
DOI: https://doi.org/10.11568/kjm.2019.27.3.735

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Published: Sep 30, 2019

Keywords:

C^{*}

-algebra, Hilbert space, m-isometry, preserving linear map

Mathematics Subject Classification:

15A86, 46L05

Alireza Majidi

Department of Mathematics Islamic Azad University Mashhad Branch, Iran

Abstract

Let $H$ be a complex Hilbert space and $B (H)$ the algebra of all bounded linear operators on $H$ . In this paper, we prove that if $φ : B (H) \to B (H)$ is a unital surjective bounded linear map, which preserves $m$ - isometries $m = 1, 2$ in both directions, then there are unitary operators $U, V \in B (H)$ such that
$\begin{array}{r} φ (T) = U T V o r φ (T) = U T^{t r} V \end{array}$
for all $T \in B (H)$ , where $T^{t r}$ is the transpose of $T$ with respect to an arbitrary but fixed orthonormal basis of $H$ .

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