Abstract. It is well-known (see [1], [9]) that the invariant vectors for a contraction $T$ on a Hilbert space ${\cal H}$ and for its adjoint $T^*$ coincide. As an immediate consequence one obtains the orthogonal decomposition $$ {\cal H}= \overline{{\cal R}(I_{\cal H}-T)} \oplus{\cal N}(I_{\cal H}-T), $$ where ${\cal N}(S)$ and ${\cal R}(S)$ denote the kernel, respectively the range, of an operator $S$ on ${\cal H}$. In this paper we obtain such a decomposition for a class of generalized contractions, more specifically for $A$-contractions, i.e. the operators $T \in{\cal B(H)}$ which satisfy $T^*AT \le A$, where $A$ is a positive operator on ${\cal H}$. We also derive several applications involving some decompositions for quasinormal operators and for oblique projections.

AMS Subject Classification (1991): 47A20, 47B20

Received January 5, 2004, and in revised form April 5, 2004. (Registered under 5843/2009.)