## 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.

### MSC:

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

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