Abstract: This article sketches a case history in which the study of composition operators does what it does best: link fundamental concepts of operator theory with the beautiful classical theory of holomorphic selfmaps of the open unit disc U. The operator-theoretic issues begin with the notion of compactness while the associated function theory revolves around Schroeder's functional equation: f(p(z))=af(z).
In Schroeder's equation the function p is the given quantity, a holomorphic self-map of the unit disc and the goal is to find a complex number a and a function f, holomorphicon U so that Schroeder's equation is satisfied.
Schroeder's equation is, of course, the eigenvalue equation for the composition operator C_p, defined by C_p f(z) = f(p(z)), where f is allowed to range through the entire space H(U) of functions holomorphic on U. This work begins with the following question:
Can you determine whether or not C_p is compact on the Hardy space H^2 by studying the growth of solutions of Schroeder's equation for p?
This article shows why the question is a natural one, both for operator theory and for classical function theory, and to sketch how it has recently been resolved. Surprisingly, it turns out that the relevant operator-theoretic concept for this problem is not compactness; it is instead the more general notion of ``Rieszness.'' The study of Riesz operators, in turn, serves as a base camp for further explorations that lead into the realm of Fredholm theory.
Download preprint (.pdf file, 194 K) to view .pdf files