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On quasi-class \(\mathcal A\) contractions. (English) Zbl 1276.47027

Let \({\mathcal Q}{\mathcal A}\) denote the class of bounded linear Hilbert space operator \(T\) which satisfy the operator inequality \(T^*|T^2|T\geq T^*|T|^2 T\). The authors prove that, if \(T\in{\mathcal Q}{\mathcal A}\) is a contraction, then either \(T\) has a nontrivial invariant subspace or \(T\) is a proper contraction and the nonnegative operator \(D= T^*(|T^2|- |T|^2)T\) is a strongly stable contraction. They extend results of [K. Takahashi and M. Uchiyama, J. Oper. Theory 10, 331–335 (1983; Zbl 0531.47010)] to completely non-normal contractions \(T\in{\mathcal Q}{\mathcal A}\) such that \(\ker T\subseteq\ker T^*\). As a generalization of [B. P. Duggal, I. H. Jeon and C. S. Kubrusly, Integral Equations Oper. Theory 49, No. 2, 141–148 (2004; Zbl 1092.47034), Theorem 3.1], they show that a commutativity theorem holds for contractions \(T\in{\mathcal Q}{\mathcal A}\). Let \(T_u\) and \(T_c\) denote the unitary part and the completely non-unitary part of \(T\), respectively. The authors extend Hastings and Wu’s results in [W. W. Hastings, Trans. Am. Math. Soc. 256, 145–161 (1979; Zbl 0381.47015)], [P. Y. Wu, Trans. Am. Math. Soc. 291, 229–239 (1985; Zbl 0582.47026)] to contractions in \({\mathcal Q}{\mathcal A}\). More precisely, if \(A= A_u\oplus A_c\) and \(B= B_u\oplus B_c\) are \({\mathcal Q}{\mathcal A}\)-contractions such that \(\mu_{A_c}<\infty\), then \(A\) and \(B\) are quasi-similar if and only if \(A_u\) and \(B_u\) are unitarily equivalent and \(A_c\) and \(B_c\) are quasi-similar.
Reviewer: Eungil Ko (Seoul)

MSC:

47B20 Subnormal operators, hyponormal operators, etc.
47A10 Spectrum, resolvent
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References:

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