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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
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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

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