Weyl’s theorem for algebraically totally hereditarily normaloid operators. (English) Zbl 1160.47301

Let \(\mathcal{L}(X)\) be the algebra of all bounded linear operators on a complex Banach space \(X\). An operator \(T\in\mathcal{L}(X)\) is said to be {normaloid} if its spectral radius equals \(\| T\| \). The operator \(T\) is said to be {hereditarily normaloid} if every part of \(T\) is normaloid (here “a part of \(T\)” means “its restriction to one of its closed invariant subspaces”), and is {totally hereditarily normaloid} if it is hereditarily normaloid and if every invertible part of \(T\) has a normaloid inverse. The class \(THN\) of totally hereditarily normaloid operators, introduced by S. V. Djordjević and the author [Math. Proc. R. Ir. Acad. 104A, 75–81 (2004; Zbl 1089.47005)], lies properly between the classes of paranormal and normaloid operators; see the recent paper by the author, S. V. Djordjević and C. S. Kubrusly [Acta Sci. Math. 71, No. 1–2, 337–352 (2005; Zbl 1106.47016)].
An operator \(T\in\mathcal{L}(X)\) is said to satisfy {property H\((q)\)} provided that \[ H_0(T-\lambda):=\{x\in X:\lim_{n\to+\infty}\| (T-\lambda)^nx\| ^{\frac{1}{n}}=0\}=\ker(T-\lambda)^q \] for all \(\lambda\in\mathbb{C}\) and some integer \(q\geq1\). The class of operators satisfying this property will be also denoted by H\((q)\). It contains, amongst others, the classes consisting of generalized scalar, subscalar and totally paranormal operators on a Banach space, multipliers of semi-simple Banach algebras, and hyponormal, \(p\)-hyponormal \((0<p<1)\) and \(M\)-hyponormal operators on a Hilbert space; see, for instance, P. Aiena [Fredholm and local spectral theory, with application to multipliers (Kluwer Acad. Press) (2004; Zbl 1077.47001)], P. Aiena and F. Villafane [Integral Equations Oper. Theory 53, No. 4, 453–466 (2005; Zbl 1097.47004)], and M. Oudghiri [Stud. Math. 163, 85–101 (2004; Zbl 1064.47004)].
Let \(\mathcal{P}\subset\mathcal{L}(X)\) be a class of operators satisfying a certain property. An operator \(T\in\mathcal{L}(X)\) is said to be {algebraically} \(\mathcal{P}\) if there exists a nonconstant complex polynomial \(p\) such that \(p(T)\in \mathcal{P}\). In this paper, the author proves that if an operator \(T\in\mathcal{L}(X)\) is algebraically H\((q)\), or \(T\) is algebraically \(THN\) and \(X\) is separable, then \(T^*\) obeys a-Weyl’s theorem, and for every function \(f\) that is analytic in an open neighborhood \(\mathcal{U}\) of the spectrum of \(T\), Weyl’s theorem holds for \(f(T)\). If, in addition, \(f\) is nonconstant on each connected component of \(\mathcal{U}\) and \(T^*\) has the single-valued extension property, then \(f(T)\) obeys a-Weyl’s theorem.


47A10 Spectrum, resolvent
47A11 Local spectral properties of linear operators
47A53 (Semi-) Fredholm operators; index theories
Full Text: DOI


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