## Roots of contractions with Hilbert-Schmidt defect operator and $$C_{\cdot 0}$$ 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 $$|ZA-(O_m\oplus N\oplus U)Z|_1= |0\oplus|X||_1$$. 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:

 47B20 Subnormal operators, hyponormal operators, etc. 47A45 Canonical models for contractions and nonselfadjoint linear operators