Paranormal contractions and invariant subspaces

B. P. Duggal; C. S. Kubrusly; N. Levan

Journal of the
Korean Mathematical Society JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2003; 40(6): 933-942

Printed November 1, 2003

Paranormal contractions and invariant subspaces

B. P. Duggal, C. S. Kubrusly, and N. Levan

United Arab Emirates University, Catholic University of Rio de Janeiro, University of California in Los Angeles

Abstract

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Abstract

It is shown that if a paranormal contraction $T$ has no nontrivial invariant subspace, then it is a proper contraction. Moreover, the nonnegative operator $Q=T^{2*}T^2-2\kern.5ptT^*T\kern-.5pt+\kern-.5ptI$ also is a proper contraction. If a quasihyponormal contraction has no nontrivial invariant subspace then, in addition, its defect operator $D$ is a proper contraction and its itself-commutator is a trace-class strict contraction. Furthermore, if one of $Q$ or $D$ is compact, then so is the other, and $Q$ and $D$ are strict contraction.

Keywords: paranormal operators, invariant subspaces, proper contractions

MSC numbers: Primary 47A15; Secondary 47B20