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Roots of contractions with Hilbert-Schmidt defect operator and ${C}_{·0}$ completely non-unitary part. (English) Zbl 0885.47007
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 $|ZA-\left({O}_{m}\oplus N\oplus U\right){Z|}_{1}={|0\oplus |X||}_{1}$. Here ${|·|}_{1}$ 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 $\left(0 or $k$-paranormal (with reducing normal subspaces) or reductive $\left({G}_{1}\right)$ with ${C}_{·0}$ completely non-unitary part, then $T$ is a coupling of the above type.
##### MSC:
 47B20 Subnormal operators, hyponormal operators, etc. 47A45 Canonical models for contractions and nonselfadjoint operators