ON k-QUASI-CLASS A CONTRACTIONS

Abstract

A bounded linear Hilbert space operator T is said to be k-quasi-class A operator if it satisfy the operator inequality <TEX>$T^{*k}{\mid}T^2{\mid}T^k{\geq}T^{*k}{\mid}T{\mid}^2T^k$</TEX> for a non-negative integer k. It is proved that if T is a k-quasi-class A contraction, then either T has a nontrivial invariant subspace or T is a proper contraction and the nonnegative operator <TEX>$D=T^{*k}({\mid}T^2{\mid}-{\mid}T{\mid}^2)T^k$</TEX> is strongly stable.