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.
