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The Numerical Range in Banach Algebras and Complex Functions of Exponential type

The Numerical Range in Banach Algebras and Complex Functions of Exponential type THE NUMERICAL RANGE IN BANACH ALGEBRAS AND COMPLEX FUNCTIONS OF EXPONENTIAL TYPE BfcLA BOLLOBAS Let A be a complex unital Banach algebra with dual A'. For x e A the numerical range of x is defined as V(A,x) = {f(x):feA' \\f\\=f(l) = \}, and the numerical radius is v(A, x) = sup {\rj\: r\ e V(A, x)}. Recently a number of results have been obtained about the relations among numerical radii and norms. For most of them see the notes of Bonsall and Duncan [3]. The proofs of the known relations are often complicated and not very revealing. In this note we shall tackle a rather general problem of this type. We shall show a natural method for determining \J{V(A,G(x)):V(A.x)czK}, in particular sup {v(A, G(x)) : V(A, x) c K] and sup {\\G(x)\\:V(A, x)c:K}, where K is an arbitrary compact convex set in the complex plane and G(z) is regular in a disc containing K and with centre the origin. It will be clear from the considerations that there is a close connection between extremal problems for numerical ranges and the theory of entire functions of expo- nential type. We shall point out that S. N. Bernstein's classical theorem http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

The Numerical Range in Banach Algebras and Complex Functions of Exponential type

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/3.1.27
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Abstract

THE NUMERICAL RANGE IN BANACH ALGEBRAS AND COMPLEX FUNCTIONS OF EXPONENTIAL TYPE BfcLA BOLLOBAS Let A be a complex unital Banach algebra with dual A'. For x e A the numerical range of x is defined as V(A,x) = {f(x):feA' \\f\\=f(l) = \}, and the numerical radius is v(A, x) = sup {\rj\: r\ e V(A, x)}. Recently a number of results have been obtained about the relations among numerical radii and norms. For most of them see the notes of Bonsall and Duncan [3]. The proofs of the known relations are often complicated and not very revealing. In this note we shall tackle a rather general problem of this type. We shall show a natural method for determining \J{V(A,G(x)):V(A.x)czK}, in particular sup {v(A, G(x)) : V(A, x) c K] and sup {\\G(x)\\:V(A, x)c:K}, where K is an arbitrary compact convex set in the complex plane and G(z) is regular in a disc containing K and with centre the origin. It will be clear from the considerations that there is a close connection between extremal problems for numerical ranges and the theory of entire functions of expo- nential type. We shall point out that S. N. Bernstein's classical theorem

Journal

Bulletin of the London Mathematical Society – Wiley

Published: Mar 1, 1971

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