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Spectrum of interpolated operators


Authors: Ernst Albrecht and Vladimir Müller
Journal: Proc. Amer. Math. Soc. 129 (2001), 807-814
MSC (2000): Primary 46B70, 47A10
DOI: https://doi.org/10.1090/S0002-9939-00-05862-7
Published electronically: September 20, 2000
MathSciNet review: 1804050
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Abstract:

Let $(X_0,X_1)$ be a compatible pair of Banach spaces and let $T$ be an operator that acts boundedly on both $X_0$ and $X_1$. Let $T_{[\theta]} \quad(0\le\theta\le 1)$ be the corresponding operator on the complex interpolation space $(X_0,X_1)_{[\theta]}$.

The aim of this paper is to study the spectral properties of $T_{[\theta]}$. We show that in general the set-valued function $\theta\mapsto \sigma(T_{[\theta]})$ is discontinuous even in inner points $\theta\in(0,1)$ and show that each operator satisfies the local uniqueness-of-resolvent condition of Ransford. Further we study connections with the real interpolation method.


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  • [A] E. Albrecht, Spectral interpolation, in: Operator Theory: Advances and Applications 14, 13-37, Birkhäuser, Basel, 1984. MR 86j:46071
  • [AS] E. Albrecht, K. Schindler, Spectrum of operators on real interpolation spaces, preprint.
  • [Au] B. Aupetit, Primer on spectral theory, Springer-Verlag, 1991. MR 92c:46001
  • [BL] J. Bergh, J. Löfström, Interpolation spaces, Springer-Verlag, 1976. MR 58:2349
  • [BKS] Yu. A. Brudnyi, S.G. Krein, E.M. Semenov, Interpolation of linear operators, Itogi nauki i tekhniki, Seriya Matematicheskii Analiz 24 (1986), 3-164. MR 88e:46056
  • [C] A.P. Calderon, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113-190. MR 29:5097
  • [F] M.K. Fort, Points of continuity of semi-continuous functions, Publicationes mathematicae Debrecen 2 (1951-52), 100-102. MR 13:764e
  • [K] M. Krause, Fredholm theory of interpolation morphisms, Recent progress in operator theory (Regensburg 1995), 219 - 231, Oper. Theory Adv. Appl. 103, Birkhäuser, Basel, 1998. MR 99h:46136
  • [R] T.J. Ransford, The spectrum of an interpolated operator and analytic multivalued functions, Pacific J. Math. 121 (1986), 445-466. MR 87c:46078
  • [S] K. Saxe, On complex interpolation and spectral continuity, Studia Math. 130 (1998), no. 3, 223-229. MR 99d:46099
  • [Sl1] Z. S\lodkowski, Analytic set-valued functions and spectra, Math. Ann. 256 (1981), 363-386. MR 83b:46070
  • [Sl2] Z. S\lodkowski, A generalization of Vesentini and Wermer's theorems, Rend. Sem. Mat. Univ. Padova, Vol. 75 (1986), 157-171. MR 88a:46091
  • [Sv] I. Ya. Sneiberg, Spectral properties of linear operators in interpolation families of Banach spaces, Mat. Issled. 9 (1974), 214-229 (Russian). MR 58:30362
  • [T] H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland, Amsterdam 1978. MR 80i:46032b
  • [Z] M. Zafran, Spectral theory and interpolation of operators, J. Funct. Anal. 36 (1980), 185-204. MR 83e:47002

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Additional Information

Ernst Albrecht
Affiliation: Fachbereich Mathematik, Universität des Saarlandes, Postfach 15 11 50, D–66041 Saarbrücken, Germany
Email: ernstalb@math.uni-sb.de

Vladimir Müller
Affiliation: Institut of Mathematics AV ČR, Zitna 25, 115 67 Prague 1, Czech Republic
Email: muller@math.cas.cz

DOI: https://doi.org/10.1090/S0002-9939-00-05862-7
Keywords: Spectrum of interpolated operators, uniqueness-of-resolvent property
Received by editor(s): September 25, 1998
Received by editor(s) in revised form: May 14, 1999
Published electronically: September 20, 2000
Additional Notes: The second author was supported by the Alexander von Humboldt Foundation and partially by grant no. 201/96/0411 of GA ČR
Communicated by: David R. Larson
Article copyright: © Copyright 2000 American Mathematical Society

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