Canonical decompositions induced by A-contractions


The classical Nagy-Foia\c s-Langer decomposition of an ordinary contraction is generalized in the context of the operators T on a complex Hilbert space \mathcal{H} which, relative to a positive operator A on \mathcal{H}, satisfy the inequality T^*AT \le A. As a consequence, a version of the classical von Neumann-Wold decomposition for isometries is derived in this context. Also one shows that, if T^*AT=A and AT=A^{1/2}TA^{1/2}, then the decomposition of \mathcal{H} in normal part and pure part relative to A^{1/2}T is just a von Neumann-Wold type decomposition for A^{1/2}T, which can be completely described.  As applications, some facts on the quasi-isometries recently studied in [4], [5], are obtained.

DOI Code: 10.1285/i15900932v28n2p187

Keywords: A-contraction; A-isometry; quasi-isometry; von Neumann-Wold decomposition

Classification: 47A15; 47A63; 47B20

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