COMPOSITION OPERATORS ON THE HARDY SPACE
SIVARAM K. NARAYAN
Let U be the open unit disk in the complex plane, let H(U) be the
space of analytic functions on U, and let H 2 (U) be the classical Hardy
space, consisting those functions in H(U) whose Maclaurin coefficients
are square summable. For φ an analytic selfmap of U, let Cφ be the
composition operator induced by φ so that Cφ f = f ◦ φ for any f ∈
H(U). Clearly Cφ preserves H(U). Littlewood proved that Cφ also
preserves H 2 (U) and thus, by the closed-graph theorem, Cφ : H 2 (U) →
H 2 (U) is a bounded linear operator.
The goal of the subject is to relate operator theoretic properties of
Cφ to the function theoretic properties of the inducing map φ. In this
introductory talk to the subject we will attempt to show this connection.
Department of Mathematics, Central Michigan University, Mount
Pleasant, Michigan 48859, USA.
E-mail address: [email protected]
1