#
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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