Duggal, B. P. Roots of contractions with Hilbert-Schmidt defect operator and \(C_{\cdot 0}\) completely non-unitary part. (English) Zbl 0885.47007 Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 36, 85-106 (1996). 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. Cited in 5 Documents MSC: 47B20 Subnormal operators, hyponormal operators, etc. 47A45 Canonical models for contractions and nonselfadjoint linear operators Keywords:controller; separable complex Hilbert space; coupling of a normal and a \(C_{10}\) 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 PDFBibTeX XMLCite \textit{B. P. Duggal}, Ann. Soc. Math. Pol., Ser. I, Commentat. Math. 36, 85--106 (1996; Zbl 0885.47007)