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    Bachir, A. Lombarkia, F. and Segres, A. 2012. Norm for Sums of Two Basic Elementary Operators. International Journal of Mathematics and Mathematical Sciences, Vol. 2012, Issue. , p. 1.
    Bachir, A. and Segres, A. 2009. Numerical range and orthogonality in normed spaces. Filomat, Vol. 23, Issue. 1, p. 21.
    Rhodius, Adolf 1976. Der numerische Wertebereich für nicht notwendig lineare Abbildungen in lokalkonvexen Räumen. Mathematische Nachrichten, Vol. 72, Issue. 1, p. 169.
    Mocanu, GH. 1974. On the numerical radius of an element of a normed algebra. Glasgow Mathematical Journal, Vol. 15, Issue. 01, p. 90.
    Bollobás, Béla 1971. The power inequality on Banach spaces. Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 69, Issue. 03, p. 411.
    Sims, Brailey 1971. A characterization of Banach-star-algebras by numerical range. Bulletin of the Australian Mathematical Society, Vol. 4, Issue. 02, p. 193.
    Duncan, J. 1969. The evaluation functionals associated with an algebra of bounded operators. Glasgow Mathematical Journal, Vol. 10, Issue. 01, p. 73.
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The numerical range of an element of a normed algebra

  • F. F. Bonsall (a1)
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Given a normed linear space X, let S(X), X′, B(X) denote respectively the unit sphere {x: ∥x∥ = 1} of X, the dual space of X, and the algebra of all bounded linear mappings of X into X. For each x ∊ S(X) and T ∊ B(X), let Dx(x) = {f e X′:∥f∥ = f(x)= 1}, and V(T; x) = {f(Tx):f∊Dx(x)}. The numerical range V(T) is then defined by

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Copyright
COPYRIGHT: © Glasgow Mathematical Journal Trust 1969
References
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1.Bohnenblust, H. F. and Karlin, S., Geometrical properties of the unit sphere of Banach algebras, Ann. of Math. 62 (1955), 217–229.
2.Bonsall, F. F., Cain, B. E., and Schneider, H., The numerical range of a continuous mapping of a normed space, Aequationes Mathematical, to appear.
3.Bonsall, F. F. and Duncan, J., Dually irreducible representations of Banach algebras, Quart. J. Math. Oxford Ser. (2) 19 (1968), 97–110.
4.Duncan, J., The evaluation functional associated with an algebra of bounded operators, Glasgow Math. J. 10 (1969), 73–76.
5.Glickfeld, B. W., On an inequality in Banach algebra geometry and semi-inner product spaces, Notices Amer. Math. Soc. 15 (1968), 339–340.
6.Holmes, R. B., A formula for the spectral radius of an operator, Amer. Math. Monthly 75 (1968) 163–166.
7.Lumer, G., Semi-inner-product spaces, Trans. Amer. Math. Soc. 100 (1961), 29–43.
8.Stone, M. H., Linear transformations in Hilbert space and their applications to analysis, Amer. Math. Soc. Col. Publ. XV (New York, 1932).
9.Williams, J. P., Spectra of products and numerical ranges, J. Math. Anal. Appl. 17 (1967), 214–220.
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Glasgow Mathematical Journal
  • ISSN: 0017-0895
  • EISSN: 1469-509X
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