On the axiomatic theory of spectrum

V. Kordula; V. Müller

Studia Mathematica (1996)

  • Volume: 119, Issue: 2, page 109-128
  • ISSN: 0039-3223

Abstract

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There are a number of spectra studied in the literature which do not fit into the axiomatic theory of Żelazko. This paper is an attempt to give an axiomatic theory for these spectra, which, apart from the usual types of spectra, like one-sided, approximate point or essential spectra, include also the local spectra, the Browder spectrum and various versions of the Apostol spectrum (studied under various names, e.g. regular, semiregular or essentially semiregular).

How to cite

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Kordula, V., and Müller, V.. "On the axiomatic theory of spectrum." Studia Mathematica 119.2 (1996): 109-128. <http://eudml.org/doc/216289>.

@article{Kordula1996,
abstract = {There are a number of spectra studied in the literature which do not fit into the axiomatic theory of Żelazko. This paper is an attempt to give an axiomatic theory for these spectra, which, apart from the usual types of spectra, like one-sided, approximate point or essential spectra, include also the local spectra, the Browder spectrum and various versions of the Apostol spectrum (studied under various names, e.g. regular, semiregular or essentially semiregular).},
author = {Kordula, V., Müller, V.},
journal = {Studia Mathematica},
keywords = {axiomatic theory of spectrum; local spectrum; semiregular operators; Browder spectrum; Apostol spectrum},
language = {eng},
number = {2},
pages = {109-128},
title = {On the axiomatic theory of spectrum},
url = {http://eudml.org/doc/216289},
volume = {119},
year = {1996},
}

TY - JOUR
AU - Kordula, V.
AU - Müller, V.
TI - On the axiomatic theory of spectrum
JO - Studia Mathematica
PY - 1996
VL - 119
IS - 2
SP - 109
EP - 128
AB - There are a number of spectra studied in the literature which do not fit into the axiomatic theory of Żelazko. This paper is an attempt to give an axiomatic theory for these spectra, which, apart from the usual types of spectra, like one-sided, approximate point or essential spectra, include also the local spectra, the Browder spectrum and various versions of the Apostol spectrum (studied under various names, e.g. regular, semiregular or essentially semiregular).
LA - eng
KW - axiomatic theory of spectrum; local spectrum; semiregular operators; Browder spectrum; Apostol spectrum
UR - http://eudml.org/doc/216289
ER -

References

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  1. [1] C. Apostol, The reduced minimum modulus, Michigan Math. J. 32 (1985), 279-294. 
  2. [2] F. F. Bonsall and J. Duncan, Complete Normed Algebras, Springer, Berlin, 1973. Zbl0271.46039
  3. [3] R. E. Curto and A. T. Dash, Browder spectral systems, Proc. Amer. Math. Soc. 103 (1988), 407-413. Zbl0662.47003
  4. [4] N. Dunford, Survey of the theory of spectral operators, Bull. Amer. Math. Soc. 64 (1958), 217-274. Zbl0088.32102
  5. [5] B. Gramsch and D. Lay, Spectral mapping theorems for essential spectra, Math. Ann. 192 (1971), 17-32. Zbl0203.45601
  6. [6] J. D. Gray, Local analytic extensions of the resolvent, Pacific J. Math. 27 (1968), 305-324. Zbl0172.17204
  7. [7] T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1966. 
  8. [8] V. Kordula, The essential Apostol spectrum and finite dimensional perturbations, to appear. Zbl0880.47005
  9. [9] V. Kordula and V. Müller, The distance from the Apostol spectrum, to appear. Zbl0861.47008
  10. [10] M. Mbekhta, Résolvant généralisé et théorie spectrale, J. Operator Theory 21 (1989), 69-105. Zbl0694.47002
  11. [11] M. Mbekhta et A. Ouahab, Opérateur s-régulier dans un espace de Banach et théorie spectrale, Publ. Inst. Rech. Math. Av. Lille 22 (1990), XII. 
  12. [12] M. Mbekhta et A. Ouahab, Contribution à la théorie spectrale généralisée dans les espaces de Banach, C. R. Acad. Sci. Paris 313 (1991), 833-836. Zbl0742.47001
  13. [13] V. Müller, On the regular spectrum, J. Operator Theory 31 (1994), 363-380. Zbl0845.47005
  14. [14] V. Rakočević, Generalized spectrum and commuting compact perturbations, Proc. Edinburgh Math. Soc. 36 (1993), 197-209. Zbl0794.47003
  15. [15] T. J. Ransford, Generalized spectra and analytic multivalued functions, J. London Math. Soc. 29 (1984), 306-322. Zbl0508.46036
  16. [16] P. Saphar, Contributions à l'étude des aplications linéaires dans un espace de Banach, Bull. Soc. Math. France 92 (1964), 363-384. Zbl0139.08502
  17. [17] C. Schmoeger, Ein Spektralabbildungssatz, Arch. Math. (Basel) 55 (1990), 484-489. 
  18. [18] C. Schmoeger, Relatively regular operators and a spectral mapping theorem, J. Math. Anal. Appl. 175 (1993), 315-320. Zbl0781.47009
  19. [19] Z. Słodkowski and W. Żelazko, On joint spectra of commuting families of operators, Studia Math. 50 (1974), 127-148. Zbl0306.47014
  20. [20] F.-H. Vasilescu, Analytic functions and some residual spectral properties, Rev. Roumaine Math. Pures Appl. 15 (1970), 435-451. Zbl0194.44101
  21. [21] F.-H. Vasilescu, Spectral mapping theorem for the local spectrum, Czechoslovak Math. J. 30 (1980), 28-35. Zbl0437.47003
  22. [22] P. Vrbová, On local spectral properties of operators in Banach spaces, ibid. 23 (1973), 483-492. Zbl0268.47006
  23. [23] W. Żelazko, Axiomatic approach to joint spectra I, Studia Math. 64 (1979), 249-261. Zbl0426.47002

Citations in EuDML Documents

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  1. J. Koliha, M. Mbekhta, V. Müller, Pak Poon, Corrigendum and addendum: "On the axiomatic theory of spectrum II"
  2. M. Berkani, Dagmar Medková, A note on the index of B -Fredholm operators
  3. Angel Martínez Meléndez, Antoni Wawrzyńczyk, An approach to joint spectra
  4. M. Mbekhta, V. Müller, On the axiomatic theory of spectrum II
  5. Vladimír Müller, Axiomatic theory of spectrum III: semiregularities
  6. M. Berkani, Restriction of an operator to the range of its powers
  7. Jacobus J. Grobler, Heinrich Raubenheimer, Andre Swartz, The index for Fredholm elements in a Banach algebra via a trace II
  8. M. Berkani, N. Castro, S. V. Djordjević, Single valued extension property and generalized Weyl’s theorem
  9. Vladimír Kordula, Vladimír Müller, Vladimir Rakočević, On the semi-Browder spectrum

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