Orthogonality preserving property for pairs of operators on Hilbert $C^*$-modules

Abstract

We investigate the orthogonality preserving property for pairs of operators on inner product $C^*$-modules. Employing the fact that the $C^*$-valued inner product structure of a Hilbert $C^*$-module is determined essentially by the module structure and by the orthogonality structure, pairs of linear and local orthogonality-preserving operators are investigated, not a priori bounded. We obtain that if ${\mathscr {A}}$ is a $C^{*}$-algebra and $T, S:{\mathscr {E}}\rightarrow {\mathscr {F}}$ are two bounded ${{\mathscr {A}}}$-linear operators between full Hilbert ${\mathscr {A}}$-modules, then $\langle x, y\rangle = 0$ implies $\langle T(x), S(y)\rangle = 0$ for all $x, y\in {\mathscr {E}}$ if and only if there exists an element $\gamma$ of the center $Z(M({{\mathscr {A}}}))$ of the multiplier algebra $M({{\mathscr {A}}})$ of ${{\mathscr {A}}}$ such that $\langle T(x), S(y)\rangle = \gamma \langle x, y\rangle$ for all $x, y\in {\mathscr {E}}$. Varying the conditions on the operators T and S we obtain further affirmative results for local operators and for pairs of a bounded and an unbounded ${{\mathscr {A}}}$-linear operator with bounded inverse.