## 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
Full Text:

### References:

  Aiena, P.; Monsalve, O., The single valued extension property and the generalized Kato decomposition property, Acta sci. math. (Szeged), 67, 461-477, (2001)  P. Aiena, F. Villafañe, Weyl’s theorem of some classes of operators, Preprint, 2003  P. Aiena, T.L. Miller, M.M. Neumann, On a localized single-valued extension property, Math. Proc. Roy. Irish Acad., in press · Zbl 1089.47004  Caradus, S.R.; Pfaffenberger, W.E.; Bertram, Y., Calkin algebras and algebras of operators on Banach spaces, (1974), Dekker New York · Zbl 0299.46062  Curto, R.E.; Han, Y.M., Weyl’s theorem, a-Weyl’s theorem and local spectral theory, J. London math. soc., 67, 499-509, (2003) · Zbl 1063.47001  Curto, R.E.; Han, Y.M., Weyl’s theorem for algebraically paranormal operators, Integral equations operator theory, 47, 307-314, (2003) · Zbl 1054.47018  Duggal, B.P.; Djordjević, S.V., Weyl’s theorem in the class of algebraically p-hyponormal operators, Comment. math. prace mat., 40, 49-56, (2000) · Zbl 0997.47019  B.P. Duggal, S.V. Djordjević, Generalized Weyl’s theorem for a class of operators satisfying a norm condition, Math. Proc. Roy. Irish Acad., in press  Duggal, B.P.; Djordjević, S.V., Weyl’s theorem through finite ascent property, Bol. soc. mat. mexicana, 10, 139-147, (2004) · Zbl 1085.47043  B.P. Duggal, S.V. Djordjević, C.S. Kubrusly, Hereditarily normaloid contractions, Acta Sci. Math. (Szeged), submitted for publication · Zbl 1106.47016  Han, Y.M.; Lee, W.Y., Weyl’s theorem holds for algebraically hyponormal operators, Proc. amer. math. soc., 128, 2291-2296, (2000) · Zbl 0953.47018  Harte, R.E.; Lee, W.Y., Another note on Weyl’s theorem, Trans. amer. math. soc., 349, 2115-2124, (1997) · Zbl 0873.47001  Heuser, H.G., Functional analysis, (1982), Wiley  Kato, T., Perturbation theory for nullity, deficiency and other quantities of linear operators, J. math. anal., 6, 261-322, (1958) · Zbl 0090.09003  Koliha, J.J., Isolated spectral points, Proc. amer. math. soc., 124, 3417-3424, (1996) · Zbl 0864.46028  Lee, W.H.; Lee, W.Y., A spectral mapping theorem for the Weyl spectrum, Glasgow math. J., 38, 61-64, (1996) · Zbl 0869.47017  Laursen, K.B.; Neumann, M.N., Introduction to local spectral theory, (2000), Clarendon Oxford · Zbl 0957.47004  Mbekhta, M., Généralisation de la décomposition de Kato aux opérateurs paranormaux et spectraux, Glasgow math. J., 29, 159-175, (1987) · Zbl 0657.47038  M. Oudghiri, Weyl’s and Browder’s theorem for operators satisfying the SVEP, Studia Math., in press · Zbl 1064.47004  Rakočević, V., Approximate point spectrum and commuting compact perturbations, Glasgow math. J., 28, 193-198, (1986) · Zbl 0602.47003  Rakočević, V., Operators obeying a-Weyl’s theorem, Rev. roumaine math. pures appl., 34, 915-919, (1989) · Zbl 0686.47005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.