A condition for spectral continuity of positive elements
HTML articles powered by AMS MathViewer
- by S. Mouton PDF
- Proc. Amer. Math. Soc. 137 (2009), 1777-1782 Request permission
Abstract:
Let $a$ be an element of a Banach algebra $A$. We introduce a compact subset $T(a)$ of the complex plane, show that the function which maps $a$ onto $T(a)$ is upper semicontinuous and use this fact to provide a condition on $a$ which ensures that if $(a_n)$ is a sequence of positive elements converging to $a$, then the sequence of the spectral radii of the terms $a_n$ converges to the spectral radius of $a$ in the case that $A$ is partially ordered by a closed and normal algebra cone and $a$ is a positive element.References
- Bernard Aupetit, A primer on spectral theory, Universitext, Springer-Verlag, New York, 1991. MR 1083349, DOI 10.1007/978-1-4612-3048-9
- Laura Burlando, Continuity of spectrum and spectral radius in Banach algebras, Functional analysis and operator theory (Warsaw, 1992) Banach Center Publ., vol. 30, Polish Acad. Sci. Inst. Math., Warsaw, 1994, pp. 53–100. MR 1285600
- Laura Burlando, Noncontinuity of spectrum for the adjoint of an operator, Proc. Amer. Math. Soc. 128 (2000), no. 1, 173–182. MR 1625705, DOI 10.1090/S0002-9939-99-05044-3
- John B. Conway and Bernard B. Morrel, Operators that are points of spectral continuity, Integral Equations Operator Theory 2 (1979), no. 2, 174–198. MR 543882, DOI 10.1007/BF01682733
- Slaviša V. Djordjević and Young Min Han, Browder’s theorems and spectral continuity, Glasg. Math. J. 42 (2000), no. 3, 479–486. MR 1793814, DOI 10.1017/S0017089500030147
- Slavisǎ V. Djordjević and Young Min Han, Spectral continuity for operator matrices, Glasg. Math. J. 43 (2001), no. 3, 487–490. MR 1878591, DOI 10.1017/S0017089501030105
- Paul Richard Halmos, A Hilbert space problem book, 2nd ed., Encyclopedia of Mathematics and its Applications, vol. 17, Springer-Verlag, New York-Berlin, 1982. MR 675952, DOI 10.1007/978-1-4684-9330-6
- H. du T. Mouton and S. Mouton, Domination properties in ordered Banach algebras, Studia Math. 149 (2002), no. 1, 63–73. MR 1881716, DOI 10.4064/sm149-1-4
- S. Mouton, A spectral problem in ordered Banach algebras, Bull. Austral. Math. Soc. 67 (2003), no. 1, 131–144. MR 1962967, DOI 10.1017/S0004972700033591
- S. Mouton, Convergence properties of positive elements in Banach algebras, Math. Proc. R. Ir. Acad. 102A (2002), no. 2, 149–162. MR 1961634, DOI 10.3318/PRIA.2002.102.2.149
- S. Mouton, On spectral continuity of positive elements, Studia Math. 174 (2006), no. 1, 75–84. MR 2239814, DOI 10.4064/sm174-1-6
- S. Mouton, On the boundary spectrum in Banach algebras, Bull. Austral. Math. Soc. 74 (2006), no. 2, 239–246. MR 2260492, DOI 10.1017/S0004972700035681
- S. Mouton and H. Raubenheimer, More spectral theory in ordered Banach algebras, Positivity 1 (1997), no. 4, 305–317. MR 1660397, DOI 10.1023/A:1009717500980
- Gerard J. Murphy, Continuity of the spectrum and spectral radius, Proc. Amer. Math. Soc. 82 (1981), no. 4, 619–621. MR 614889, DOI 10.1090/S0002-9939-1981-0614889-2
- J. D. Newburgh, The variation of spectra, Duke Math. J. 18 (1951), 165–176. MR 51441, DOI 10.1215/S0012-7094-51-01813-3
- H. Raubenheimer and S. Rode, Cones in Banach algebras, Indag. Math. (N.S.) 7 (1996), no. 4, 489–502. MR 1620116, DOI 10.1016/S0019-3577(97)89135-5
- Helmut Schaefer, Some spectral properties of positive linear operators, Pacific J. Math. 10 (1960), 1009–1019. MR 115090, DOI 10.2140/pjm.1960.10.1009
Additional Information
- S. Mouton
- Affiliation: Department of Mathematical Sciences, University of Stellenbosch, Private Bag X1, Matieland 7602, South Africa
- Email: smo@sun.ac.za
- Received by editor(s): June 29, 2007
- Received by editor(s) in revised form: April 22, 2008, and July 21, 2008
- Published electronically: November 4, 2008
- Additional Notes: The author thanks the referee for making useful suggestions.
- Communicated by: N. Tomczak-Jaegermann
- © Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 137 (2009), 1777-1782
- MSC (2000): Primary 46H05, 47A10
- DOI: https://doi.org/10.1090/S0002-9939-08-09715-3
- MathSciNet review: 2470837