Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [A] Ernst Albrecht, Spectral interpolation, Spectral theory of linear operators and related topics (Timişoara/Herculane, 1983) Oper. Theory Adv. Appl., vol. 14, Birkhäuser, Basel, 1984, pp. 13–37. MR 789606
  • [AS] E. Albrecht, K. Schindler, Spectrum of operators on real interpolation spaces, preprint.
  • [Au] Bernard Aupetit, A primer on spectral theory, Universitext, Springer-Verlag, New York, 1991. MR 1083349
  • [BL] Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. MR 0482275
  • [BKS] Yu. A. Brudnyĭ, S. G. Kreĭn, and E. M. Semënov, Interpolation of linear operators, Mathematical analysis, Vol. 24 (Russian), Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1986, pp. 3–163, 272 (Russian). Translated in J. Soviet Math 42 (1988), no. 6, 2009–2112. MR 887950
  • [C] A.-P. Calderón, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. MR 0167830
  • [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) Oper. Theory Adv. Appl., vol. 103, Birkhäuser, Basel, 1998, pp. 219–231. MR 1635025
  • [R] T. J. Ransford, The spectrum of an interpolated operator and analytic multivalued functions, Pacific J. Math. 121 (1986), no. 2, 445–466. MR 819200
  • [S] Karen Saxe, On complex interpolation and spectral continuity, Studia Math. 130 (1998), no. 3, 223–229. MR 1625210
  • [Sl1] Zbigniew Słodkowski, Analytic set-valued functions and spectra, Math. Ann. 256 (1981), no. 3, 363–386. MR 626955, https://doi.org/10.1007/BF01679703
  • [Sl2] Zbigniew Slodkowski, A generalization of Vesentini and Wermer’s theorems, Rend. Sem. Mat. Univ. Padova 75 (1986), 157–171. MR 847664
  • [Sv] I. Ja. Šneĭberg, Spectral properties of linear operators in interpolation families of Banach spaces, Mat. Issled. 9 (1974), no. 2(32), 214–229, 254–255 (Russian). MR 0634681
  • [T] H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. MR 500580
    Hans Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library, vol. 18, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503903
  • [Z] Misha Zafran, Spectral theory and interpolation of operators, J. Funct. Anal. 36 (1980), no. 2, 185–204. MR 569253, https://doi.org/10.1016/0022-1236(80)90099-3

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 46B70, 47A10

Retrieve articles in all journals with MSC (2000): 46B70, 47A10


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