Let \(A\) be a fixed nonzero positive bounded linear operator on a complex Hilbert space \(\mathcal{H}\). Then a bounded linear operator \(T\) on \(\mathcal{H}\) is said to be an \(A\)-contraction if \(T^* A T \leq A\). The operator \(T\) is said to be an \(A\)-isometry if \(T^*AT = A\). Notice that \(T\) is a contraction if and only if it is an \(I\)-contraction, where \(I\) is the identity operator on \(\mathcal{H}\).
In the paper under review, the author analyzes the largest invariant subspace of an \(A\)-contraction on which the restriction is an \(A\)-isometry. Under some additional hypotheses, the author obtains some results on the existence of invariant subspaces. This theory applies to the class of quasinormal contractions. In particular, the author obtains the unitary, isometric and quasi-isometric parts of a quasinormal contraction.