The Spectral Radius of a Hermitian Element of a Banach Algebra

Abstract

The following remarkable theorem was recently proved by Sinclair [3], THEOREM.The spectral radius of a Hermitian element of a complex unital Banach algebra is equal to its norm.Our purpose is to give an elementary proof of Sinclair's theorem. This will make available an elementary proof of the Vidav-Palmer characterization of C*-algebras [4,2].Let A be a complex unital Banach algebra. The numerical range V(a) of an element a e A is given by V(a) = {f(a): / e D(l)}, where D(\) denotes the set of all continuous linear functionals f on A such that

Power-Bounded Operators and Related Norm Estimates

“…To illustrate the ideas, let us first give a continuous time version. The methods used are similar to those in a paper by Bonsall and Crabb [7] in their proof of a special case of Sinclair's theorem [28]. After this present paper was finished, the authors learned of the papers by Berkani, Esterle and Mokhtari [4] and Esterle and Mokhtari [14] which use similar methods.…”

“…For example we have the following corollaries. The next theorem is a generalization of the argument used by Bonsall and Crabb [7] to prove a special case of Sinclair's theorem [28], namely that the norm of a hermitian element A of a Banach algebra coincides with its spectral radius r(A). Theorem 4.11.…”

On the norm and numerical radius of Hermitian elements

“…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.…”

The Norm of a Hermitian Element in a Banach Algebra

“…The proof uses a generalisation of Bernstein's theorem which gives a bound on the derivative of an entire function along the real line. F. F. Bonsall and M. J. Crabb [2] have recently given an elementary proof of our Proposition 2 when ß is zero (which is equivalent to ß real). In Lemma 5 and Proposition 6 we construct a norm on the algebra of polynomials, in one indeterminate x, which is maximal with respect to the property that x is hermitian of norm one.…”