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 Contractions of class $\mathcal Q$ and invariant subspaces Bull. Korean Math. Soc. 2005 Vol. 42, No. 1, 169-177 Published online March 1, 2005 B. P. Duggal, C. S. Kubrusly, and N. Levan , Catholic University of Rio de Janeiro, University of California in Los Angeles Abstract : A Hilbert Space operator $T$ is of class $\Q$ if $T^{2*}T^2-2\kern1ptT^*T+I$ is nonnegative. Every paranormal operator is of class $\Q$, but class-$\Q$ operators are not necessarily normaloid. It is shown that if a class-$\Q$ contraction $T$ has no nontrivial invariant subspace, then it is a proper contraction. Moreover, the nonnegative operator $Q=T^{2*}T^2-2\kern1ptT^*T+I$ also is a proper contraction. Keywords : paranormal operators, invariant subspaces, proper contractions MSC numbers : Primary 47A15; Secondary 47B20 Downloads: Full-text PDF