Norms of linear-fractional composition operators

Abstract

We obtain a representation for the norm of the composition operator C Φ on the Hardy space H 2 whenever Φ is a linear-fractional mapping of the form Φ(z) = b/(cz + d). The representation shows that, for such mappings Φ, the norm of C Φ always exceeds the essential norm of C Φ . Moreover, it shows that a formula obtained by Cowen for the norms of composition operators induced by mappings of the form Φ(z) = sz + t has no natural generalization that would yield the norms of all linear-fractional composition operators. For rational numbers s and t, Cowen's formula yields an algebraic number as the norm; we show, e.g., that the norm of C 1/(2-z) is a transcendental number. Our principal results are based on a process that allows us to associate with each non-compact linear-fractional composition operator C Φ , for which ∥C Φ ∥ > ∥C Φ ∥ e , an equation whose maximum (real) solution is ∥C Φ ∥ 2 . Our work answers a number of questions in the literature; for example, we settle an issue raised by Cowen and MacCluer concerning co-hyponormality of a certain family of composition operators.