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Zbl 0885.47007
Duggal, B.P.
Roots of contractions with Hilbert-Schmidt defect operator and $C\sb{\cdot 0}$ completely non-unitary part. (English)
[J] Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 36, 85-106 (1996). ISSN 0373-8299

Summary: Let $T$ be a controller on a separable complex Hilbert space such that $T$ is a coupling of a normal and a $C_{10}$ contraction. If $A$ is an $m$th root of $T$, where $A$ has Hilbert-Schmidt defect operator, then there exists a nilpotent operator $O_m$ acting on a finite-dimensional Hilbert space, a normal contraction $N$, a unilateral shift $U$, a quasi-affinity $Z$ and an operator $X$ of trace class such that $|ZA-(O_m\oplus N\oplus U)Z|_1= |0\oplus|X||_1$. Here $|\cdot|_1$ denotes the trace norm. If also the spectrum of $A$ is a subset of the reals, then $A$ is similar to the direct sum of a nilpotent $O_m$ and a self-adjoint contraction $M$. It is shown that if a contraction $T$ has Hilbert-Schmidt defect operator and is either dominant or injective $k$-quasihyponormal or $p$-hyponormal $(0<p<1)$ or $k$-paranormal (with reducing normal subspaces) or reductive $(G_1)$ with $C_{\cdot 0}$ completely non-unitary part, then $T$ is a coupling of the above type.

MSC 2000:: *47B20 Subnormal operators, etc.
47A45 Canonical models for contractions and nonselfadjoint operators

Keywords: controller; separable complex Hilbert space; coupling of a normal and a $C\sb{10}$ contraction; Hilbert-Schmidt defect operator; nilpotent operator; unilateral shift; quasi-affinity; trace class; injective $k$-quasihyponormal; $p$-hyponormal; $k$-paranormal; reducing normal subspaces; completely non-unitary part

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Zentralblatt MATH Berlin [Germany]