Composition operators and Schroeder's functional equation

 

Joel H. Shapiro

 

Contemp. Math. 213 (1998) 213--228

 

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.

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