Canonical decompositions induced by $A$-contractions

Abstract

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.