Norms of linear-fractional composition operators

Abstract:

We obtain a representation for the norm of the composition operator $C_\phi$ on the Hardy space $H^2$ whenever $\phi$ is a linear-fractional mapping of the form $\phi (z) = b/(cz +d)$. The representation shows that, for such mappings $\phi$, the norm of $C_\phi$ always exceeds the essential norm of $C_\phi$. Moreover, it shows that a formula obtained by Cowen for the norms of composition operators induced by mappings of the form $\phi (z) = sz +t$ 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 $C_\phi$, for which $\|C_\phi \|> \|C_\phi \|_e$, an equation whose maximum (real) solution is $\|C_\phi \|^2$. 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.