Extremal perturbations of semi-Fredholm operators

Thorsten Kröncke

Studia Mathematica (1998)

  • Volume: 129, Issue: 3, page 253-264
  • ISSN: 0039-3223

Abstract

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Let T be a bounded operator on an infinite-dimensional Banach space X and Ω a compact subset of the semi-Fredholm domain of T. We construct a finite rank perturbation F such that min[dim N(T+F-λ), codim R(T+F-λ)] = 0 for all λ ∈ Ω, and which is extremal in the sense that F² = 0 and rank F = max{min[dim N(T-λ), codim R(T-λ)] : λ ∈ Ω.

How to cite

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Kröncke, Thorsten. "Extremal perturbations of semi-Fredholm operators." Studia Mathematica 129.3 (1998): 253-264. <http://eudml.org/doc/216503>.

@article{Kröncke1998,
abstract = {Let T be a bounded operator on an infinite-dimensional Banach space X and Ω a compact subset of the semi-Fredholm domain of T. We construct a finite rank perturbation F such that min[dim N(T+F-λ), codim R(T+F-λ)] = 0 for all λ ∈ Ω, and which is extremal in the sense that F² = 0 and rank F = max\{min[dim N(T-λ), codim R(T-λ)] : λ ∈ Ω.},
author = {Kröncke, Thorsten},
journal = {Studia Mathematica},
keywords = {semi-Fredholm operators; minimum index; extremal perturbations; semi-Fredholm domain; finite rank operators},
language = {eng},
number = {3},
pages = {253-264},
title = {Extremal perturbations of semi-Fredholm operators},
url = {http://eudml.org/doc/216503},
volume = {129},
year = {1998},
}

TY - JOUR
AU - Kröncke, Thorsten
TI - Extremal perturbations of semi-Fredholm operators
JO - Studia Mathematica
PY - 1998
VL - 129
IS - 3
SP - 253
EP - 264
AB - Let T be a bounded operator on an infinite-dimensional Banach space X and Ω a compact subset of the semi-Fredholm domain of T. We construct a finite rank perturbation F such that min[dim N(T+F-λ), codim R(T+F-λ)] = 0 for all λ ∈ Ω, and which is extremal in the sense that F² = 0 and rank F = max{min[dim N(T-λ), codim R(T-λ)] : λ ∈ Ω.
LA - eng
KW - semi-Fredholm operators; minimum index; extremal perturbations; semi-Fredholm domain; finite rank operators
UR - http://eudml.org/doc/216503
ER -

References

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  1. [Bo88] Z. Boulmaarouf, The Laffey-West decomposition, Proc. Roy. Irish Acad. Sect. A 88 (1988), 125-131. Zbl0675.47008
  2. [Bo90] Z. Boulmaarouf, Décomposition de Laffey-West globale, thèse, l'Université de Nice, 1990. 
  3. [FöJa92] K.-H. Förster and K. Jahn, Extremal compressions and perturbations of bounded operators, ibid. 92 (1992), 289-296. Zbl0791.47006
  4. [FöKr96] K.-H. Förster and M. Krause, On extremal perturbations of semi-Fredholm operators, Integral Equations Operator Theory 26 (1996), 125-135. Zbl0864.47003
  5. [Is87] G. G. Islamov, Control of the spectrum of a dynamical system, Differential Equations 23 (1987), 872-875. Zbl0681.93038
  6. [Ka58] T. Kato, Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261-322. Zbl0090.09003
  7. [Ka66] T. Kato, Perturbation Theory for Linear Operators, Springer, New York, 1966. 
  8. [Kr95] T. Kröncke, Extremale Störungen von Fredholmoperatoren, Diplomarbeit, Technische Universität Berlin, 1995. 
  9. [LaWe82] T. J. Laffey and T. T. West, Fredholm commutators, Proc. Roy. Irish Acad. Sect. A 87 (1987), 137-146. 
  10. [MaSe78] A. S. Markus and A. A. Sementsul, Operators which weakly perturb the spectrum, Sibirsk. Mat. Zh. 19 (1978), 646-653 (in Russian). 
  11. [Mb93] M. Mbekhta, Semi-Fredholm perturbations and commutators, Math. Proc. Cambridge Philos. Soc. 113 (1993), 173-177. 
  12. [Ó S88] M. Ó Searcóid, Economical finite rank perturbations of semi-Fredholm operators, Math. Z. 198 (1988), 431-434. 
  13. [Sł86] Z. Słodkowski, Operators with closed ranges in spaces of analytic vector-valued functions, J. Funct. Anal. 69 (1986), 155-177. Zbl0611.46046
  14. [We90] T. T. West, Removing the jump-Kato's decomposition, Rocky Mountain J. Math. 20 (1990), 603-612. Zbl0726.47006
  15. [Ze92] J. Zemánek, An analytic Laffey-West decomposition, Proc. Roy. Irish Acad. Sect. A 92 (1992), 101-106. Zbl0785.47008

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