Advertisement

manuscripta mathematica

, Volume 2, Issue 2, pp 191–202 | Cite as

Spektrum, numerischer Wertebereich und IHRE Maximumprinzipien in Banachalgebren

  • Bernd Schmidt
Article

Abstract

By a function of support a numerical range is defined for elements of complex Banach algebras. The geometric character of the definition allows to proof some results about the numerical range in a very natural way.- In the second part holomorphic perturbations of the numerical range and the spectrum are treated as perturbations of the function of support to get a maximum principle for the numerical range and the convex hull of the spectrum, whereas there is in general no maximum principle for the spectrum. This answers a question raised by A.Brown and R.G.Douglas.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Literatur

  1. [1]
    BOHNENBLUST, H.F. a. S.KARLIN: Geometrical properties of the unit sphere of Banach algebras. Ann. Math.(2) 62, 217–229 (1955).Google Scholar
  2. [2]
    BONNESEN, T. u. W.FENCHEL: Theorie der konvexen Körper. Berlin: Springer 1934.Google Scholar
  3. [3]
    BROWN, A. a. R.G.DOUGLAS: On maximum theorems for analytic operator functions. Acta Sci. Math.(Szeged) 26, 325–327 (1965).Google Scholar
  4. [4]
    DOUGLAS, R.G. a. P.ROSENTHAL: A necessary and sufficient condition that an operator be normal. J. Math. Anal. Appl. 22, 10–11 (1968).Google Scholar
  5. [5]
    EMBRY, M.R.: Conditions implying normality in Hilbert space. Pacific J. Math. 18, 457–460 (1966).Google Scholar
  6. [6]
    GLICKFELD, B.W.: On an inequality in Banach algebra geometry and semi inner product spaces. Notices Amer. Math. Soc. 15, 654–12, S.339 (1968).Google Scholar
  7. [7]
    HILDEBRANDT, S.: The closure of the numerical range of an operator as spectral set. Comm. Pure Appl. Math. 17, 415–421 (1964).Google Scholar
  8. [8]
    HILDEBRANDT, S.: Über den numerischen Wertebereich eines Operators. Math. Ann. 163, 230–247 (1966).Google Scholar
  9. [9]
    HILLE, E. a. R.S.PHILLIPS: Functional analysis and semi-groups. Revised edition, Providence, R.I. Amer. Math. Soc. 1957.Google Scholar
  10. [10]
    KANTOROVITZ, S.: On the characterization of spectral operators. Trans. Amer. Math. Soc. 111, 152–181 (1964).Google Scholar
  11. [11]
    KRASNOSELSKII, M.A.: A class of linear operators in the space of abstract continuous functions. Math. Notes 2 (1967), 856–858 (1968).Google Scholar
  12. [12]
    LUMER, G.: Semi-inner-product-spaces. Trans. Amer. Math. Soc. 100, 29–43 (1961).Google Scholar
  13. [13]
    ORLAND, G.: On a class of operators. Proc. Amer. Math. Soc. 15, 75–80 (1964).Google Scholar
  14. [14]
    PUTNAM, C.R.: Commutation properties of Hilbert space operators. Berlin-Heidelberg-New York: Springer 1967.Google Scholar
  15. [15]
    RADÓ, T.: Subharmonic functions. Berlin:Springer 1937.Google Scholar
  16. [16]
    VESENTINI, E.: On the subharmonicity of the spectral radius. Boll.Un.mat.Ital.IV.Ser. 1, 427–429 (1968)Google Scholar
  17. [17]
    VIDAV, I.: Eine metrische Kennzeichnung der selbstadjungierten Operatoren. Math.Z. 66, 121–128 (1956)Google Scholar
  18. [18]
    WILLIAMS, J.P.: Spectra of products and numerical ranges. J. Math. Anal. Appl. 17, 214–220 (1967).Google Scholar
  19. [19]
    ŻELAZKO, W.: A characterization of multiplicative linear functionals in complex Banach algebras. Studia Math. 30, 83–85 (1968).Google Scholar

Copyright information

© Springer-Verlag 1970

Authors and Affiliations

  • Bernd Schmidt
    • 1
  1. 1.Mathematisches Institut der Universität65 Mainz

Personalised recommendations