Geodesic
Polynomials in a hermitian element
Glasgow mathematical journal, Tome 30 (1988) no. 2, pp. 171-176

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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]. 
Crabb, M. J.; McGregor, C. M. Polynomials in a hermitian element. Glasgow mathematical journal, Tome 30 (1988) no. 2, pp. 171-176. doi: 10.1017/S0017089500007187
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