Vol. 44, No. 1, 1973

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Linear maps of the disk algebra

Richard Rochberg

Vol. 44 (1973), No. 1, 337–354

Denote by A the disk algebra, the sup-normed Banach algebra of functions continuous on the closed unit disk of the complex plane and analytic in the interior of the disk. This paper describes the elements of K(F,G), the set of linear maps of A into itself which are of norm one, fix the constants, and take the inner function F to the inner function G. It is shown that any L in K(F,G) is determined by its action on a certain complement of FA and that the image of this complement under L must be orthogonal (in H2) to ZGA. These facts are used to show that K(F,G) is a compact convex set of real dimension at most (n 1)(m + 1) where n and m are the orders of F and G respectively. This result gives examples of non-multiplicative extreme points in the set of linear maps of A into itself which are of norm one and fix the constants. Some analysis is made of K(F,G) when F and G are not required to be inner.

Mathematical Subject Classification 2000
Primary: 46J15
Received: 23 August 1971
Published: 1 January 1973
Richard Rochberg
Department of Mathematics
Washington University in St. Louis
Campus Box 1146
One Brookings Dr
Saint Louis MO 63130-4899
United States