Norms of linear-fractional composition operators. (English) Zbl 1038.47500

Summary: 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 C. C. Cowen [Integral Equations Oper. Theory 11, 151–160 (1988; Zbl 0638.47027)] 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 C. C. Cowen and B. D. MacCluer [Contemp. Math. 213, 17–25 (1998; Zbl 0908.47025)] concerning co-hyponormality of a certain family of composition operators.


47B33 Linear composition operators
Full Text: DOI


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