On the norm and numerical radius of Hermitian elements

Abstract

Let A be a complex Banach algebra with a unit 1. We denote by Sp(a) the spectrum of an element a E A, and by [a[ = max{lz[: z ~ Sp(a)} its spectral radius. The compact convex set V(a) = {~0(a): ~o A*, I1~011 = ~o(1) = 1}, where A* denotes the adjoint space of A, is called the numerical range of a, and the number v(a) = max{Izl: z ~ V(a)} is called the numerical radius ofa. An element a ~ A is called Hermitian if II exp(ita)ll = 1 for all t ~ ~. This is equivalent to the requirement V(a) C R [8]; the latter implies, in particular, that the spectrum of an Hermitian a is real. In the case of the algebra of bounded operators in a Hilbert space the Hermite property is equivalent to the self-adjointness. The obvious inequalities lal ~< v(a) ~< Ilall become equalities for an Hermitian a: this is established by Vidav [13] for the first equality, and by Katsnelson [3] and independently by Bonsall and Crabb [7] for the second one. Such coincidences were also indicated for functions F of an Hermitian a. We consider some cases of another equality( 1) Below a is always assumed to be an Hermitian element with Sp(a) = [-or, or], cr > 0. The conditions on F implying (1) dearly depend on the norm of the algebra (a) generated by a. For the minimal norm equal to the spectral radius, (a) is a C*-algebra, and then (1) is satisfied for any continuous F. The maximal norm on (a) generates another extremal algebra denoted by E~ [-cr, or]. It has several realizations. Following the one presented in [10,12], E~ [-a, cr] consists of the functions F admitting on [-or, a] the representation [-cr, cr], for any element v of the algebra M(II~) of finite complex Borel measures on ]g the element f)(t) = f~ exp(itx)dr(x) of Ea [-<r, or] is correctly defined. Consequently, if Z(cr) is a closed-in-M(IR) ideal of all measures having Fourier-Stieltjes transformations vanishing on [-or, or], then Ea [-~r, or] is isomorphic to the factor-algebra Mfig)/f.(cr); hence below we identify the latter with E~ [-~r, a]. The measures of the coset of b ~ Ea [-O, cr] are called the measures related to b. To describe the adjoint space of E~ [-cr, or] one can use, for example, the local isomorphy of the Fourier algebras for L10R) and M(IR) (cf., e.g., [2]), which implies that Ea [-cr, or] is also isomorphic to the algebra Lt(IR)/ZlflR), where ZICR) is an analogous ideal of the functions from LI(~) that have Fourier transformations vanishing on [-or, cr]. This allows us to identify (E~ [-cr, or])* with a certain subspace of Loo(~) which is easily determined as the set B~ of all entire functions Vilnius University,