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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 mth 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 NU)Z| 1 =|0|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 (0<p<1) or k-paranormal (with reducing normal subspaces) or reductive (G 1 ) with C ·0 completely non-unitary part, then T is a coupling of the above type.
47B20Subnormal operators, hyponormal operators, etc.
47A45Canonical models for contractions and nonselfadjoint operators