# CONTRACTIONS OF CLASS Q AND INVARIANT SUBSPACES

• DUGGAL, B.P. (5 Tudor Court, Amherst Road, London W13 8NE, England) ;
• KUBRUSLY, C.S. (Catholic University of Rio de Janeiro) ;
• LEVAN, N. (University of California Los Angeles, Los Angeles)
• Published : 2005.02.01

#### Abstract

A Hilbert Space operator T is of class Q if $T^2{\ast}T^2-2T{\ast}T + I$ 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 = $T^2{\ast}T^2-2T{\ast}T + I$ also is a proper contraction.

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