Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-15T11:38:48.649Z Has data issue: false hasContentIssue false
  • English
  • Français

Polynomials in a hermitian element

Published online by Cambridge University Press:  18 May 2009

M. J. Crabb  and
C. M. McGregor
M. J. Crabb
Affiliation:
Department of Mathematics, University of Glasgow
C. M. McGregor
Affiliation:
Department of Mathematics, University of Glasgow
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For an element a of a unital Banach algebra A with dual space A′, we define the numerical range V(a) = {f(a):f ∊ A′, ∥f∥ = f(1) = 1}, and the numerical radius v(a) = sup{⃒z⃒:z ∊ V(a)}. An element a is said to be Hermitian if V(a) ⊆ ℝ ,equivalently ∥exp (ita)∥ = 1(t ∊ ℝ). Under the condition V(h) ⊆ [-1, 1], any polynomial in h attains its greatest norm in the algebra Ea[-1,1], generated by an element h with V(h) = [-1, 1].

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 30 , Issue 2 , May 1988 , pp. 171 - 176
Copyright
Copyright © Glasgow Mathematical Journal Trust 1988

References

1.Baillet, M., Un calcul fonctionnel de class C 1 pour les operateurs pre-hermitiens d'un espace de Banach, C.R. Acad. Sci. Paris, Ser A, 283 (1976), 891–894.Google Scholar
2.Boas, R. P., Entire functions (Academic Press, 1954).Google Scholar
3.Crabb, M. J. and McGregor, C. M., Numerical ranges of powers of Hermitian elements, Glasgow Math. J. 28 (1986), 37–45.CrossRefGoogle Scholar
4.Konig, H., A functional calculus for Hermitian elements of complex Banach algebras, Archiv der Math. 28 (1977), 422–430.CrossRefGoogle Scholar
5.Sinclair, A. M., The Banach algebra generated by a Hermitian operator, Proc. London Math. Soc. (3) 24 (1972), 681–691.Google Scholar
6.Sinclair, A. M., The Banach algebra generated by a derivation, Operator Theory: Adv. Appl. 14 (1984), 241–250.Google Scholar
7.Szego, G., Orthogonal polynomials, American Math. Soc. Coll. Publ. XXIII (1959).Google Scholar