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)
Perturbation of spectra of operator matrices and local spectral theory. (English) Zbl 1105.47006
Let $X$ and $Y$ be Banach spaces and let $L(X,Y)$ denote the space of all bounded linear operators from $X$ to $Y$ ($L(X):=L(X,X)$). For $A\in L(X)$, $B\in L(Y)$ and $C\in L(X,Y)$, denote by $M_{C}$ the operator defined on $X\oplus Y$ by $\left[\smallmatrix A & C\\ 0 & B \endsmallmatrix \right]$. In this article, the defect set $D^{\Sigma }=(\Sigma (A)\cup \Sigma (B))\setminus \Sigma (M_{C})$ is studied for different spectra including the spectrum, the essential spectrum, the Weyl spectrum and the approximate point spectrum. The obtained results are applied to the stability of such spectra ($D^{\Sigma }=\varnothing$) and the classes of operators $C$ for which stability holds of $M_{C}$ using local spectral theory.

47A11Local spectral properties
47A55Perturbation theory of linear operators
47A10Spectrum and resolvent of linear operators
Full Text: DOI
[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
[2] Albrecht, E.; Eschmeier, J.: Analytic functional models and local spectral theory. Proc. London math. Soc. (3) 75, 323-348 (1997) · Zbl 0881.47007
[3] Barraa, M.; Boumazgour, M.: A note on the spectrum of an upper triangular operator matrix. Proc. amer. Math. soc. 131, 3083-3088 (2003) · Zbl 1050.47005
[4] Bishop, E.: A duality theorem for an arbitrary operator. Pacific J. Math. 9, 379-397 (1959) · Zbl 0086.31702
[5] 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
[6] Djordjević, D. S.: Perturbations of spectra of operator matrices. J. operator theory 48, 467-486 (2002) · Zbl 1019.47003
[7] Djordjević, S. V.; Han, Y. M.: A note on Weyl’s theorem for operator matrices. Proc. amer. Math. soc. 130, 2543-2547 (2003) · Zbl 1041.47006
[8] Du, H. K.; Pan, J.: Perturbation of spectrums of $2\times 2$ operator matrices. Proc. amer. Math. soc. 121, 761-776 (1994) · Zbl 0814.47016
[9] Elbjaoui, H.; Zerouali, E. H.: Local spectral theory for $2\times 2$ operator matrices. Int. J. Math. math. Sci. 42, 2667-2672 (2003) · Zbl 1060.47003
[10] Finch, J. K.: The single valued extension property on a Banach space. Pacific J. Math. 58, 61-69 (1975) · Zbl 0315.47002
[11] Han, J. K.; Lee, H. Y.; Lee, W. Y.: Invertible completions of $2\times 2$ upper triangular operator matrices. Proc. amer. Math. soc. 129, 119-123 (2000) · Zbl 0944.47004
[12] Houimdi, M.; Zguitti, H.: Propriétés spectrales locales d’une matrice carrée des opérateurs. Acta math. Vietnam 25, 137-144 (2000)
[13] Hwang, I. S.; Lee, W. Y.: The boundedness below of $2\times 2$ upper triangular operator matrices. Integral equations operator theory 39, 267-276 (2001) · Zbl 0986.47004
[14] K.B. Laursen, M.M. Neumann, On analytic solutions of the equation (T - \lambda )f(\lambda )=x, in: LEU Seminar Notes in Funct. Anal. & PDEs, 1993 -- 1994, pp. 256 -- 265
[15] Laursen, K. B.; Neumann, M. M.: An introduction to local spectral theory. (2000) · Zbl 0957.47004
[16] Lee, W. Y.: Weyl spectra for operator matrices. Proc. amer. Math. soc. 129, 131-138 (2000) · Zbl 0965.47011
[17] Oberai, K. K.: On the weyls spectrum (II). Illinois. J. Math. 21, 84-90 (1977) · Zbl 0358.47004
[18] Pearcy, C. M.: Some recent developments in operator theory. Cbms 36 (1978) · Zbl 0444.47001
[19] Schweinsberg, A.: The operator equation AX - XB=C with normal A and B. Pacific J. Math. 102, 447-453 (1982) · Zbl 0524.47012
[20] Schweinsberg, A.: Similarity orbits and the range of the generalized derivations X$\to $MX - XN. Trans. amer. Math. soc. 324, 201-211 (1991) · Zbl 0722.47033
[21] Stampfli, J.: Hyponormal operators. Pacific J. Math. 12, 1453-1458 (1962) · Zbl 0129.08701
[22] Zerouali, E. H.; Zguitti, H.: On the weak decomposition property $(\delta w)$. Studia math. 167, 17-28 (2005) · Zbl 1202.47005