Essential spectra through local spectral theory

Laursen, K.

doi:10.1090/S0002-9939-97-03852-5

Essential spectra through local spectral theory
HTML articles powered by AMS MathViewer

by K. B. Laursen PDF

Proc. Amer. Math. Soc. 125 (1997), 1425-1434 Request permission

Abstract:

Based on a nice observation of Eschmeier, this is a study of the use of local spectral theory in investigations of the semi-Fredholm spectrum of a continuous linear operator. We also examine the retention of the semi-Fredholm spectrum under weak intertwining relations; it is shown, inter alias, that if two decomposable operators are intertwined asymptotically by a quasi-affinity then they have identical semi-Fredholm spectra. The results are applied to multipliers on commutative semisimple Banach algebras.

References

Pietro Aiena, Riesz multipliers on commutative semisimple Banach algebras, Arch. Math. (Basel) 54 (1990), no. 3, 293–303. MR 1037620, DOI 10.1007/BF01188526
Pietro Aiena and Kjeld B. Laursen, Multipliers with closed range on regular commutative Banach algebras, Proc. Amer. Math. Soc. 121 (1994), no. 4, 1039–1048. MR 1185257, DOI 10.1090/S0002-9939-1994-1185257-1
E Albrecht, J Eschmeier: Analytic functional models and local spectral theory (manuscript, 1987)
E. Albrecht and R. D. Mehta, Some remarks on local spectral theory, J. Operator Theory 12 (1984), no. 2, 285–317. MR 757436
Ion Colojoară and Ciprian Foiaş, Theory of generalized spectral operators, Mathematics and its Applications, Vol. 9, Gordon and Breach Science Publishers, New York-London-Paris, 1968. MR 0394282
J Eschmeier, K B Laursen, M M Neumann: Multipliers with natural local spectra on commutative Banach algebras, J Functional Analysis, to appear.
Domingo A. Herrero, On the essential spectra of quasisimilar operators, Canad. J. Math. 40 (1988), no. 6, 1436–1457. MR 990108, DOI 10.4153/CJM-1988-066-x
Ronald Larsen, An introduction to the theory of multipliers, Die Grundlehren der mathematischen Wissenschaften, Band 175, Springer-Verlag, New York-Heidelberg, 1971. MR 0435738
K B Laursen: Spectral subspaces and automatic continuity, Doctoral Dissertation, Copenhagen 1991.
K. B. Laursen and M. Mbekhta, Closed range multipliers and generalized inverses, Studia Math. 107 (1993), no. 2, 127–135. MR 1244571
Kjeld B. Laursen, Vivien G. Miller, and Michael M. Neumann, Local spectral properties of commutators, Proc. Edinburgh Math. Soc. (2) 38 (1995), no. 2, 313–329. MR 1335876, DOI 10.1017/S0013091500019106
Kjeld B. Laursen and Michael M. Neumann, Local spectral properties of multipliers on Banach algebras, Arch. Math. (Basel) 58 (1992), no. 4, 368–375. MR 1152625, DOI 10.1007/BF01189927
K. B. Laursen and M. M. Neumann, Asymptotic intertwining and spectral inclusions on Banach spaces, Czechoslovak Math. J. 43(118) (1993), no. 3, 483–497. MR 1249616
Kjeld B. Laursen and Michael M. Neumann, Local spectral theory and spectral inclusions, Glasgow Math. J. 36 (1994), no. 3, 331–343. MR 1295508, DOI 10.1017/S0017089500030937
K. B. Laursen and P. Vrbová, Some remarks on the surjectivity spectrum of linear operators, Czechoslovak Math. J. 39(114) (1989), no. 4, 730–739. MR 1018009
Mostafa Mbekhta, Généralisation de la décomposition de Kato aux opérateurs paranormaux et spectraux, Glasgow Math. J. 29 (1987), no. 2, 159–175 (French). MR 901662, DOI 10.1017/S0017089500006807
Mostafa Mbekhta, Local spectrum and generalized spectrum, Proc. Amer. Math. Soc. 112 (1991), no. 2, 457–463. MR 1045142, DOI 10.1090/S0002-9939-1991-1045142-X
T L Miller, V G Miller: Equality of essential spectra of quasisimilar operators with property $(\delta )$, Glasgow Math. J, to appear.
V G Miller, M M Neumann: Local spectral theory for multipliers and convolution operators, in Algebraic methods in operator theory, Birkhäuser, Boston, 1994.
M M Neumann: Local spectral theory for operators on Banach space and applications to convolution operators on group algebras, in Seminar Notes in Functional Analysis and PDEs 1993/94, Department of Mathematics, Louisiana State University, Baton Rouge, LA, 1994.
Ch. Schmoeger, Ein Spektralabbildungssatz, Arch. Math. (Basel) 55 (1990), no. 5, 484–489 (German). MR 1079997, DOI 10.1007/BF01190270
Christoph Schmoeger, On isolated points of the spectrum of a bounded linear operator, Proc. Amer. Math. Soc. 117 (1993), no. 3, 715–719. MR 1111438, DOI 10.1090/S0002-9939-1993-1111438-8
T. T. West, A Riesz-Schauder theorem for semi-Fredholm operators, Proc. Roy. Irish Acad. Sect. A 87 (1987), no. 2, 137–146. MR 941708
Misha Zafran, On the spectra of multipliers, Pacific J. Math. 47 (1973), 609–626. MR 326309