The power inequality on normed spaces

Abstract

Let X be a complex normed space, with dual space X′. Let T be a bounded linear operator on X. The numerical range V(T) of T is defined as {f(Tx): x ∈ X, f ∈ X′, ‖ x ‖ = ‖ f ‖ = f(x) = 1}, and the numerical radius v(T) of T is defined as sup {|z|: z ∈ V(T)}. For a unital Banach algebra A, the numerical range V(a) of a ∈ A is defined as V(Ta), where Ta is the operator on A defined by Tab = ab. It is shown in (2, Chapter 1.2, Lemma 2) that V(a) = {f(a): f ∈ D(1)}, where D(1) = {f ∈ A′: ‖f‖ = f(1) = 1}.