Inequalities for the numerical radius in unital normed algebras

Abstract In this paper, some inequalities between the numerical radius of an element from a unital normed algebra and certain semi-inner products involving that element and the unity are given.


Introduction
Let A be a unital normed algebra over the complex number field C and let a ∈ A. Recall that the numerical radius of a is given by (see [2, p. 15 It is known that v(·) is a norm on A that is equivalent to the given norm · . More precisely, the following double inequality holds: for any a ∈ A, where e = exp (1). Following [2], we notice that this crucial result appears slightly hidden in Bohnenblust and Karlin [1,Theorem 1] together with the inequality x ≤ eΨ(x), where Ψ(x) = sup{|λ| −1 log e λx } over λ complex, λ = 0, which occurs on page 219. A simpler proof was given by Lumer [5], though with the constant 1/4 in place of 1/e. For a simple proof of (1.2) that borrows ideas from Lumer and from Glickfeld [6], see [2, p. 34].
A generalisation of (1.2) for powers has been obtained by M. J. Crabb [3] who proved that for any a ∈ A.
In this paper, some inequalities between the numerical radius of an element and the superior semi-inner product of that element and the unity in the normed algebra A are given via the celebrated representation result of Lumer from [5].

Some subsets in
For λ ∈ C and r > 0, we define the subset of A bȳ The following result holds. Proposition 1. Let λ ∈ C and r > 0. Then∆(λ, r) is a closed convex subset of A and The following representation result may be stated.
Proposition 2. For any γ, Γ ∈ C, γ = Γ, we have: Proof. We observe that for any z ∈ C we have the equivalence This follows by the equality that holds for any z ∈ C.
The equality (2.2) is thus a simple conclusion of this fact.
Making use of some obvious properties in C and for continuous linear functionals, we can state the following corollary as well.
Now, if we assume that Re(Γ ) ≥ Re(γ) and Im(Γ ) ≥ Im(γ), then we can define the following subset of A : One can easily observe thatS(γ, Γ ) is closed, convex and

Semi-inner products and Lumer's theorem
Let (X, · ) be a normed linear space over the real of complex number field K. The mapping f : X → R, f (x) = 1 2 x 2 is obviously convex and then there exist the following limits: x, y s = lim t→0 + y + tx 2 − y 2 2t for every two elements x, y ∈ X. The mapping ·, · s ( ·, · i ) will be called the superior semi-inner product (the interior semi-inner product) associated to the norm · .
The following result essentially due to Lumer [5] (see [2, p. 17]) can be stated. Theorem 1. Let A be a unital normed algebra over K (K = C, R). For each a ∈ A, where V (a) is the numerical range of a (see for instance [2, p. 15]).
Remark 1. In terms of semi-inner products, the above identity can be stated as: The following result that provides more information may be stated.
Theorem 2. For any a ∈ A, we have: 2t is the superior semi-inner product associated with the numerical radius.

Proof.
Since v(a) ≤ a , we have: Now, let f ∈ D(1). Then, for each α > 0, Taking the infimum over α > 0, we deduce that If we now take the supremum over f ∈ D(1) in (3.4), we obtain: which, by Lumer's identity, implies that a, 1 s ≤ a, 1 v,s .
Corollary 2. The following inequality holds Proof. Schwarz's inequality for the norm v(.) gives that and by (3.3), the inequality (3.5) is proved.

Reverse inequalities for the numerical radius
Utilising the inequality (3.5) we observe that for any complex number β located in the closed disc centered in 0 and with radius 1 we have | βa, 1 s | as a lower bound for the numerical radius v(a). Therefore, it is a natural question to ask how far these quantities are from each other under various assumptions for the element a in the unital normed algebra A and the scalar β. A number of results answering this question are incorporated in the following theorems.
The following result may be stated as well.