×

Linear orthogonality preservers of standard operator algebras. (English) Zbl 1227.47024

Let \(\theta : A\to B\) be a linear surjective mapping, where \(A\) and \(B\) are two standard operator algebras on (real or complex) Hilbert spaces \(H\) and \(K\), respectively. In this paper, the authors give a unified approach to characterize the following linear range/domain orthogonality preservers:
(1) \(ab=0\Leftrightarrow \theta (a)\theta (b)=0\);
(2) \(a^*b=0\Leftrightarrow \theta(a)^*\theta (b)=0\);
(3) \(ab^*=0\Leftrightarrow \theta (a)\theta (b)^*=0\);
(4) \(a^*b=0\Leftrightarrow \theta (a)\theta (b)^*=0\);
(5) \(ab^*=0\Leftrightarrow \theta (a)^*\theta (b)=0\);
(6) \(a^*b=ab^*=0\Leftrightarrow \theta (a)^*\theta (b)=\theta (a)\theta (b)^*=0\).
They show that all these preservers carry a standard form, and are thus automatically bounded.

MSC:

47B49 Transformers, preservers (linear operators on spaces of linear operators)
47L10 Algebras of operators on Banach spaces and other topological linear spaces
PDFBibTeX XMLCite
Full Text: DOI