Journal of the
Korean Mathematical Society JKMS

It is shown that if $A \ge 0$ and $T$ are two bounded linear operators on a complex Hilbert space $\mathcal H$ satisfying the inequality $T^*AT \le A$ and the condition $AT=A^{1/2}TA^{1/2}$, then there exists the maximum reducing subspace for $A$ and $A^{1/2}T$ on which the equality $T^*AT=A$ is satisfied. We concretely express this subspace in two ways, and as applications, we derive certain decompositions for quasinormal contractions. Also, some facts concerning the quasi-isometries are obtained.

Keywords: $A$-contraction, $A$-isometry, $A$-weighted isometry, quasinormal operator, quasi-isometry

MSC numbers: Primary 47A15, 47A63; Secondary 47B20