Local spectral theory and spectral inclusions

Laursen, Kjeld B. ; Neumann, Michael M.

Glasgow mathematical journal, Tome 36 (1994) no. 3, pp. 331-343

Résumé

Suppose that T and S are continuous linear operators on complex Banach spaces X and Y, respectively, and that A is a non-zero continuous linear mapping from X to Y. If A intertwines T and S in the sense that SA = AT, then a classical result due to Rosenblum implies that the spectra σ(T) and σ(S) must overlap, see [12]. Actually, Davis and Rosenthal [5]have shown that the surjectivity spectrum σsu(T) will meet the approximate point spectrum σap(S) in this case (terms to be denned below). Further information about the relations between the two spectra and their finer structure becomes available when the intertwiner A is injective or has dense range, see [9], [12], [13].

Détail

DOI : 10.1017/S0017089500030937

Bibliographie

[1] 1.Albrecht, E. and Eschmeier, J., Analytic functional models and local spectral theory (submitted). Google Scholar

[2] 2.Berberian, S. K., Lectures in functional analysis and operator theory (Springer-Verlag, 1974). Google Scholar | DOI

[3] 3.Bishop, E., A duality theorem for an arbitrary operator, Pacific J. Math. 9 (1959), 379–397. Google Scholar | DOI

[4] 4.Colojoară, I. and Foiaş, C., Theory of generalized spectral operators (Gordon and Breach, New York, 1968). Google Scholar

[5] 5.Davis, C. and Rosenthal, P., Solving linear operator equations, Canad. J. Math. 26 (1974), 1384–1389. Google Scholar | DOI

[6] 6.Engelking, R., General topology (Polish Scientific Publishers, Warsaw, 1977). Google Scholar

[7] 7.Eschmeier, J., Analytische Dualität und Tensorprodukte in der mehrdimensionalen Spektraltheorie, Habilitationsschrift, Schriftenreihe Math. Inst. Universität Münster 2.42, 1987. Google Scholar

[8] 8.Eschmeier, J. and Prunaru, B., Invariant subspaces for operators with Bishop's property (β) and thick spectrum, J. Functional Analysis 94 (1990), 196–222. Google Scholar | DOI

[9] 9.Fialkow, L. A., A note on quasisimilarity of operators, Ada Sci. Math. (Szeged) 39 (1977), 67–85. Google Scholar

[10] 10.Foiaş, C., and Vasilescu, F.-H., On the spectral theory of commutators, J. Math. Anal. Appl. 31 (1970), 473–486. Google Scholar | DOI

[11] 11.Frunză, S., A complement to the duality theorem for decomposable operators, Rev. Roumaine Math. Pures Appl. 28 (1983), 475–478. Google Scholar

[12] 12.Grabiner, S., Spectral consequences of the existence of intertwining operators, Comment. Math. Pace Mat. 22 (1980/1981), 227–238. Google Scholar

[13] 13.Hoover, T. B., Quasisimilarity of operators, Illinois J. Math. 16 (1972), 678–686. Google Scholar | DOI

[14] 14.Kato, T., Perturbation theory for nullity, deficiency and other quantities of linear operators, J. Analyse Math. 6 (1958), 261–322. Google Scholar | DOI

[15] 15.Laursen, K. B. and Vrbová, P.;, Some remarks on the surjectivity spectrum of linear operators, Czech. Math. J. 39 (1989), 730–739. Google Scholar | DOI

[16] 16.Laursen, K. B. and Neumann, M. M., Asymptotic intertwining and spectral inclusions on Banach spaces, Czech. Math. J. 43 (1993), 483–497. Google Scholar | DOI

[17] 17.Leiterer, J., Banach coherent analytic Fréchet sheaves, Math. Nachr. 85 (1978), 91–109. Google Scholar | DOI

[18] 18.Miller, V. G. and Neumann, M. M., Local spectral theory for multipliers and convolution operators, in Algebraic methods in operator theory (Birkhäuser-Verlag, Boston, 1994), 25–36. Google Scholar | DOI

[19] 19.Schmoeger, C., Ein Spektralabbildungssatz, Arch. Math (Basel) 55 (1990), 484–489. Google Scholar | DOI

[20] 20.Sun, S. L., The single-valued extension property and spectral manifolds, Proc. Amer. Math. Soc. 118, (1993), 77–87. Google Scholar | DOI

[21] 21.Vasilescu, F.-H., Analytic functional calculus and spectral decompositions (Editura Academiei and D. Reidel Publishing Company, Bucharest and Dordrecht, 1982). Google Scholar