Orthogonality preserving property for pairs of operators on Hilbert $C^*$-modules
Michael Frank, M. S. Moslehian, Ali Zamani
TL;DR
This paper addresses the problem of characterizing when a pair of operators on a Hilbert $C^*$-module preserves orthogonality in one direction, extending classical Wigner-type results from Hilbert spaces to modules. The main approach shows that, under standard assumptions (full modules and nonzero $\mathscr{A}$-linear operators), the orthogonality condition $x \perp y$ implying $T(x) \perp S(y)$ is equivalent to the existence of a scalar or central multiplier $\gamma$ such that $\langle T(x), S(y)\rangle = \gamma\,\langle x,y\rangle$ for all $x,y$. This yields strong structural consequences: $T$ and $S$ become $\mathscr{A}$-linear, and in bounded/invertible cases one often obtains $T=(S^*)^{-1}$ with $\gamma$ in the center $Z(M(\mathscr{A}))$, while the two-operator setting is extended via bidual techniques. A real rank zero variant shows $\gamma$ can lie in $Z(\mathscr{A})$ under a self-adjoint condition, and the paper provides examples that demonstrate limits such as non-invertible $\gamma$ and nontrivial range behavior that differ from the Hilbert-space scenario. Overall, the results generalize key orthogonality principles to the Hilbert $C^*$-module framework, clarifying how the $C^*$-algebraic and multiplier-centre structures govern two-operator orthogonality preservation and invertibility relations.
Abstract
We investigate the orthogonality preserving property for pairs of mappings on inner product $C^*$-modules extending existing results for a single orthogonality-preserving mapping. Guided by the point of view 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 mappings are investigated, not a priori bounded. The intuition is that most often $C^*$-linearity and boundedness can be derived from the settings under consideration. In particular, we obtain that if $\mathscr{A}$ is a $C^{*}$-algebra and $T, S:\mathscr{E}\longrightarrow \mathscr{F}$ are two bounded ${\mathscr A}$-linear mappings 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 $γ$ of the center $Z(M({\mathscr A}))$ of the multiplier algebra $M({\mathscr A})$ of ${\mathscr A}$ such that $\langle T(x), S(y)\rangle = γ\langle x, y\rangle$ for all $x, y\in \mathscr{E}$. In particular, for adjointable operators $S$ we have $T=(S^*)^{-1}$, and any bounded invertible module operator $T$ may appear. Varying the conditions on the mappings $T$ and $S$ we obtain further affirmative results for local operators and for pairs of a bounded and of an unbounded module operator with bounded inverse, among others. Also, unbounded operators with disjoint ranges are considered. The proving techniques give new insights.