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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
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