Norms of linear-fractional composition operators

• Published in: Transactions of the American Mathematical Society Vol. 356; no. 6; pp. 2459 - 2480
• Main Authors: Bourdon, P. S., Fry, E. E., Hammond, C., Spofford, C. H.
• Format: Journal Article
• Language: English
• Published: Providence, RI American Mathematical Society 01-06-2004
• Subjects: Abstracting; Algebra; Automorphisms; Coefficients; Exact sciences and technology; Mathematical analysis; Mathematical theorems; Mathematics; Operator theory; Polynomials; Real numbers; Sciences and techniques of general use; Transcendentals; Written composition; Transcendental number; Linear form; Linear fractional mapping; Composition operator; Cowen formula; Algebraic number; Linear operator; Standards; Operator theory; Hardy space
• ISSN: 0002-9947; 1088-6850
• DOI: 10.1090/S0002-9947-03-03374-9

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.