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A note on the Browder’s and Weyl’s theorem. (English) Zbl 1163.47005
Let $$T$$ be a bounded linear operator on a Banach space $$X$$ and let $$\sigma(T)$$, $$\sigma_w(T)$$, $$\sigma_b(T)$$, and $$\sigma_{bw}(T)$$ be its spectrum, Weyl spectrum, Browder spectrum, and B-Weyl spectrum, respectively. One says that Weyl’s theorem holds for $$T$$ if $$\sigma(T)\setminus\sigma_w(T)=E_0(T)$$, where $$E_0(T)$$ is the set of all isolated points of $$\sigma(T)$$ which are eigenvalues of finite multiplicity. Browder’s theorem holds for $$T$$ if $$\sigma_w(T)=\sigma(T)\setminus \pi_0(T)$$, where $$\pi_0(T)$$ is the set of all isolated points of $$\sigma(T)$$ for which the corresponding spectral projection is of finite rank. The generalized Weyl’s theorem says that $$\sigma(T)\setminus\sigma_{bw}(T)=E(T)$$, where $$E(T)$$ is the set of all isolated points of $$\sigma(T)$$ which are eigenvalues. Further, $$T$$ is said to satisfy the generalized Browder’s theorem if $$\sigma(T)\setminus\sigma_{bw}(T)=\pi(T)$$, where $$\pi(T)$$ is the set of all poles of the resolvent of $$T$$. Finally, one says that $$T$$ has the single-valued extension property (SVEP) at $$\lambda\in{\mathbb C}$$ if, for every open disk $$D(\lambda,r)$$, the null function is the only analytic solution of the equation $$(T-\mu)f(\mu)=0$$ for all $$\mu\in D(\lambda,r)$$.
The authors show several results relating the above defined concepts. In particular, they prove that the following properties are equivalent:
(i)
$$T$$ satisfies Browder’s theorem;
(ii)
$$T$$ has the SVEP at $$\lambda\notin\sigma_w(T)$$;
(iii)
$$T$$ satisfies generalized Browder’s theorem;
(iv)
$$T$$ has the SVEP at $$\lambda\notin\sigma_{bw}(T)$$.
Another representative result is the following. Suppose that $$E(T)=\pi(T)$$. Then the following properties are equivalent:
(a) $$T$$ satisfies Weyl’s theorem;
(b) $$T$$ satisfies Browder’s theorem;
(c) $$T$$ has the SVEP at all $$\lambda\notin\sigma_w(T)$$;
(d) $$T$$ has the SVEP at all $$\lambda\notin\sigma_{bw}(T)$$.
Some applications of the obtained results are given.

##### MSC:
 47A53 (Semi-) Fredholm operators; index theories 47A10 Spectrum, resolvent 47A11 Local spectral properties of linear operators
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##### References:
  Heuser, H. G.: Functional Analysis, John Wiley and Sons, 1982 · Zbl 0465.47001  Berkani, M., Koliha, J. J.: Weyl type theorems for bounded linear operators. Acta Sci. Math. (Szeged), 69, 359–376 (2003) · Zbl 1050.47014  Weyl, H.: Über beschränkte quadratische formen, deren Differenz vollsteig ist. Rend. Circ. Mat. Palermo, 27, 373–392 (1909) · JFM 40.0395.01 · doi:10.1007/BF03019655  Coburn, L. A.: Weyl’s theorem for nonnormal operators. Michigan Math. J., 13, 285–288 (1966) · Zbl 0173.42904 · doi:10.1307/mmj/1031732778  Berberian, S. K.: An extension of Weyl’s theorem to a class of not necessarily normal operators. Michigan Math. J., 16, 273–279 (1969) · Zbl 0175.13603 · doi:10.1307/mmj/1029000272  Berberian, S. K.: The Weyl spectrum of an operator. Indiana Univ. Math. J., 20, 529–544 (1970) · Zbl 0203.13401 · doi:10.1512/iumj.1970.20.20044  Curto, R. E., Han, Y. M.: Weyl’s theorem for algebraically paranormal operators. Integr. Equ. Oper. Theory., 47, 307–314 (2003) · Zbl 1054.47018 · doi:10.1007/s00020-002-1164-1  Duggal, B. P.: Hereditarily normaloid operators. Extracta Math., 20(2), 203–217 (2005) · Zbl 1097.47005  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 · doi:10.1090/S0002-9939-00-05741-5  Lee, W. Y.: Weyl’s theorem for operator matrices. Integral Equations and Operator Theory, 32, 319–331 (1998) · Zbl 0923.47001 · doi:10.1007/BF01203773  Oudghiri, M.: Weyl’s and Browder’s theorem for operators satisfying the SVEP. Studia Math., 163(1), 85–101 (2004) · Zbl 1064.47004 · doi:10.4064/sm163-1-5  Harte, R. E., Lee, W. Y.: Another note on Weyl’s theorem. Trans. Amer. Math. Soc., 349, 2115–2124 (1997) · Zbl 0873.47001 · doi:10.1090/S0002-9947-97-01881-3  Berkani, M.: Index of B-Fredholm operators and generalization of a Weyl theorem. Proc. Amer. Math. Soc., 130, 1717–1723 (2002) · Zbl 0996.47015 · doi:10.1090/S0002-9939-01-06291-8  Berkani, M., Arroud, A.: Generalized Weyl’s theorem and hyponormal operators. J. Aust. Math. Soc., 76, 291–302 (2004) · Zbl 1061.47021 · doi:10.1017/S144678870000896X  Cao, X., Guo, M., Meng, B.: Weyl type theorem for p-hyponormal and M-hyponormal operators. Studia Math., 163, 177–188 (2004) · Zbl 1075.47011 · doi:10.4064/sm163-2-5  Zguitti, H.: A note on generalized Weyl’s theorem. J. Math. Anal. Appl., 316, 373–381 (2006) · Zbl 1101.47002 · doi:10.1016/j.jmaa.2005.04.057  Amouch, M.: Weyl type theorems for operators satisfying the single-valued extension property. J. Math. Anal. Appl., 326, 1476–1484 (2007) · Zbl 1117.47007 · doi:10.1016/j.jmaa.2006.03.085  Amouch, M.: Generalized a-Weyl’s theorem and the single-valued extension property. Extracta. Math., 21(1), 51–65 (2006) · Zbl 1123.47005  Amouch, M., Zguitti, H.: On the equivalence of Browder’s and generalized Browder’s theorem. Glasgow Math. J., 48, 179–185 (2006) · Zbl 1097.47012 · doi:10.1017/S0017089505002971  Finch, J. K.: The single valued extension property on a Banach space. Pacific J. Math., 58, 61–69 (1975) · Zbl 0315.47002  Aiena, P., Monsalve, O.: The single valued extension property and the generalized Kato decomposition property. Acta Sci. Math. (Szegzed), 67, 461–477 (2001) · Zbl 1017.47008  Koliha, J. J.: Isolated spectral points. Proc. Amer. Math. Soc., 124, 3417–3424 (1996) · Zbl 0864.46028 · doi:10.1090/S0002-9939-96-03449-1  Aiena, P., Monsalve, O.: Operators which do not have the single valued extension property. J. Math. Anal. Appl., 250, 435–448 (2000) · Zbl 0978.47002 · doi:10.1006/jmaa.2000.6966  Dunford, N., Schwartz, J. T.: Linear Operators. Part I, Interscience, New York, 1964 · Zbl 0128.34803  Duggal, B. P.: Browder-Weyl theorems for polaroid operators satisfying an orthogonality property, the International Conference on Functional Analysis, Operator Theory and Application, FAOT 2005, Italy  Chourasia, N. N., Ramanujan, P. B.: Paranormal operators on Banach spaces. Bull. Austral. Math. Soc., 21, 161–168 (1980) · Zbl 0417.47005 · doi:10.1017/S0004972700005980  Schmoeger, C.: On operators T such that Weyl’s theorem holds for f(T). Extracta Math., 13, 27–33 (1998) · Zbl 0977.47003  Mbekhta, M.: Généralisation de la décomposition de Kato aux opérateurs paranormaux et spectraux. Glasgow Math. J., 29, 159–175 (1987) · Zbl 0657.47038 · doi:10.1017/S0017089500006807  Mbekhta, M.: Résolvant généralisé et théorie spectrale. J. Operator Theory, 21, 69–105 (1989) · Zbl 0694.47002
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