zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
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.

47A10Spectrum and resolvent of linear operators
47A53(Semi-) Fredholm operators; index theories
47A55Perturbation theory of linear operators
Full Text: DOI
[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. · Zbl 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