MAXIMUM SUBSPACES RELATED TO A-CONTRACTIONS AND QUASINORMAL OPERATORS

Abstract

It is shown that if <TEX>$A{\geq}0$</TEX> and T are two bounded linear operators on a complex Hilbert space H satisfying the inequality <TEX>$T^*\;AT{\leq}A$</TEX> and the condition <TEX>$AT=A^{1/2}TA^{1/2}$</TEX>, then there exists the maximum reducing subspace for A and <TEX>$A^{1/2}T$</TEX> on which the equality <TEX>$T^*\;AT=A$</TEX> 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.