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An abstract model for compressions. (English) Zbl 0669.47005

The authors investigate the geometry of a Hilbert space \({\mathcal K}\) containing two subspaces \({\mathcal A}\) and \({\mathcal B}\) whose sum \({\mathcal A}+{\mathcal B}\) is dense in \({\mathcal K}\). If \({\mathcal A}^{\perp}\cap {\mathcal B}^{\perp}=\{0\}\), define D:\({\mathcal K}\to {\mathcal A}^{\perp}\) by \(D=(P_{{\mathcal A}^{\perp}}P_{{\mathcal B}}P_{{\mathcal A}^{\perp}})^{1/2},\) define V:\({\mathcal A}^{\perp}\to {\mathcal B}\) by \(VD=P_{{\mathcal B}}P_{{\mathcal A}^{\perp}},\) then V is an isometry mapping \({\mathcal K}\) into the direct sum \({\mathcal A}\oplus {\mathcal B}\). If U is a bounded linear operator on \({\mathcal K}\) for which \({\mathcal A}\) and \({\mathcal B}\) are reducing subspaces, V intertwines the restrictions \(U| {\mathcal A}\) and \(U| {\mathcal B}-({\mathcal B}\cap {\mathcal A}).\) These considerations are used to obtain direct sum relationships between a contraction \(T\in {\mathcal B}({\mathcal H})\) and its minimal unitary dilation \(U\in {\mathcal B}({\mathcal K})\).
Reviewer: R.W.Shonkwiler

MSC:

47A20 Dilations, extensions, compressions of linear operators
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