Orthogonality preserving property for pairs of operators on Hilbert $C^{\ast }$-modules

Abstract

We investigate the orthogonality preserving property for pairs of mappings on inner product $C^{\ast }$-modules extending existing results for a single orthogonality-preserving mapping. Guided by the point of view that the $C^{\ast }$-valued inner product structure of a Hilbert $C^{\ast }$-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 $C^{\ast }$-linearity and boundedness can be derived from the settings under consideration. In particular, we obtain that if $\mathcal{A}$ is a $C^{\ast }$-algebra and $T,S:\mathcal{E}⟶\mathcal{F}$ are two bounded $\mathcal{A}$-linear mappings between full Hilbert $\mathcal{A}$-modules, then $⟨x,y⟩=0$ implies $⟨T(x),S(y)⟩=0$ for all $x,y\in \mathcal{E}$ if and only if there exists an element $\gamma$ of the center $Z(M(\mathcal{A}))$ of the multiplier algebra $M(\mathcal{A})$ of $\mathcal{A}$ such that $⟨T(x),S(y)⟩=\gamma ⟨x,y⟩$ for all $x,y\in \mathcal{E}$. In particular, for adjointable operators $S$ we have $T=(S^{\ast }{)}^{-1}$, 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.

23 pages, In this last revision several new examples are added and some minor changes appeared in the text. To appear in Aequat. Math

References in corpus (2)

• Orthogonality of bounded linear operators on complex Banach spaces
• On some geometric properties of operator spaces