# zbMATH — the first resource for mathematics

Weyl’s theorem for operator matrices. (English) Zbl 0923.47001
Summary: “Weyl’s theorem holds” for an operator when the complement in the spectrum of the “Weyl spectrum” coincides with the isolated points of the spectrum which are eigenvalues of finite multiplicity. By comparison “Browder’s theorem holds” for an operator when the complement in the spectrum of the Weyl spectrum coincides with Riesz points. Weyl’s theorem and Browder’s theorem are liable to fail for $$2\times 2$$ operator matrices. In this paper we explore how Weyl’s theorem and Browder’s theorem survive for $$2\times 2$$ operator matrices in Hilbert space.

##### MSC:
 47A10 Spectrum, resolvent 47A53 (Semi-) Fredholm operators; index theories 47A55 Perturbation theory of linear operators
Full Text:
##### References:
 [1] S.K. Berberian,An extension of Weyl’s theorem to a class of not necessarily normal operators, Michigan Math. J.16 (1969), 273-279. · Zbl 0175.13603 · doi:10.1307/mmj/1029000272 [2] S.K. Berberian,The Weyl spectrum of an operator, Indiana Univ. Math. J.20 (1970), 529-544. · Zbl 0203.13401 · doi:10.1512/iumj.1970.20.20044 [3] B. Chevreau,On the specral picture of an operator, J. Operator Theory4 (1980), 119-132. · Zbl 0465.47015 [4] N.N. Chourasia,On Weyl’s theorem for spectral operators and essential spectra of direct sum, Pure Appl. Math. Sci.15 (1982), 39-45. · Zbl 0505.47024 [5] L.A. Coburn,Weyl’s theorem for nonnormal operators, Michigan Math. J.13 (1966), 285-288. · Zbl 0173.42904 · doi:10.1307/mmj/1031732778 [6] R.G. Douglas,Banach Algebra Techniques in the Operator Theory, Academic press, New York, 1972. [7] Hong-Ke Du and Jin Pan,Perturbation of spectra of 2$$\times$$2 operator matrices, Proc. Amer. Math. Soc.121 (1994), 761-766. · Zbl 0814.47016 · doi:10.1090/S0002-9939-1994-1185266-2 [8] D.R. Farenick and W.Y. Lee,Hyponormality and spectra of Toeplitz operators, Trans. Amer. Math. Soc.348 (1996), 4153-4174. · Zbl 0862.47013 · doi:10.1090/S0002-9947-96-01683-2 [9] I. Gohberg, S. Goldberg and M.A. Kaashoek,Classes of Linear Operators (vol I), Birkhäuser, Basel, 1990. · Zbl 0745.47002 [10] P.R. Halmos,Hilbert Space Problem Book, Springer, New York, 1984. [11] R.E. Harte,Fredholm, Weyl and Browder theory, Proc. Royal Irish Acad.85A (2) (1985), 151-176. [12] R.E. Harte,Invertibility and Singularity for Bounded Linear Operators, Dekker, New York, 1988. · Zbl 0636.47001 [13] R.E. Harte,Invertibility and singularity of operator matrices, Proc. Royal Irish Acad.88A (2) (1988), 103-118. · Zbl 0678.47001 [14] R.E. Harte and W.Y. Lee,Another note on Weyl’s theorem, Trans. Amer. Math. Soc.349 (1997), 2115-2124. · Zbl 0873.47001 · doi:10.1090/S0002-9947-97-01881-3 [15] R.E. Harte, W.Y. Lee and L.L. Littlejohn,On generalized Riesz points (to appear). [16] W.Y. Lee,Weyl spectra of operator matrices (to appear). · Zbl 0965.47011 [17] W.Y. Lee and S.H. Lee,A spectral mapping theorem for the Weyl spectrum, Glasgow Math. J.38(1) (1996), 61-64. · Zbl 0869.47017 · doi:10.1017/S0017089500031268 [18] K.K. Oberai,On the Weyl spectrum, Illinois J. Math.18 (1974), 208-212. · Zbl 0277.47002 [19] K.K. Oberai,On the Weyl spectrum (II), Illinois J. Math.21 (1977), 84-90. · Zbl 0358.47004 [20] C.M. Pearcy,Some Recent Developements in Operator Theory, CBMS 36, Providence: AMS, 1978. · Zbl 0444.47001 [21] C. Schmoeger,Ascent, descent and the Atkinson region in Banach algebras II, Ricerche di Matematica vol.XLII, fasc.2o (1993), 249-264. · Zbl 0807.46054 [22] H. Weyl,Über beschränkte quadratische Formen, deren Differenz vollsteig ist, Rend. Circ. Mat. Palermo27 (1909), 373-392. · JFM 40.0395.01 · doi:10.1007/BF03019655 [23] H. Widom,On the spectrum of a Toeplitz operator, Pacific J. Math.14 (1964), 365-375. · Zbl 0197.10902
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.