##
**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.

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.

### MSC:

47B33 | Linear composition operators |

PDF
BibTeX
XML
Cite

\textit{P. S. Bourdon} et al., Trans. Am. Math. Soc. 356, No. 6, 2459--2480 (2004; Zbl 1038.47500)

Full Text:
DOI

### References:

[1] | Matthew J. Appel, Paul S. Bourdon, and John J. Thrall, Norms of composition operators on the Hardy space, Experiment. Math. 5 (1996), no. 2, 111 – 117. · Zbl 0862.47015 |

[2] | P. Avramidou and F. Jafari, On norms of composition operators on Hardy spaces, Function spaces (Edwardsville, IL, 1998) Contemp. Math., vol. 232, Amer. Math. Soc., Providence, RI, 1999, pp. 47 – 54. · Zbl 0938.47023 |

[3] | Alan Baker, Transcendental number theory, Cambridge University Press, London-New York, 1975. · Zbl 0297.10013 |

[4] | Paul S. Bourdon and Joel H. Shapiro, Cyclic phenomena for composition operators, Mem. Amer. Math. Soc. 125 (1997), no. 596, x+105. · Zbl 0996.47032 |

[5] | Paul S. Bourdon and Dylan Q. Retsek, Reproducing kernels and norms of composition operators, Acta Sci. Math. (Szeged) 67 (2001), no. 1-2, 387 – 394. · Zbl 1003.47017 |

[6] | Paul S. Bourdon, David Levi, Sivaram K. Narayan, and Joel H. Shapiro, Which linear-fractional composition operators are essentially normal?, J. Math. Anal. Appl. 280 (2003), no. 1, 30 – 53. · Zbl 1024.47008 |

[7] | Carl C. Cowen, Composition operators on \?², J. Operator Theory 9 (1983), no. 1, 77 – 106. · Zbl 0504.47032 |

[8] | Carl C. Cowen, Linear fractional composition operators on \?², Integral Equations Operator Theory 11 (1988), no. 2, 151 – 160. · Zbl 0638.47027 |

[9] | Carl C. Cowen and Thomas L. Kriete III, Subnormality and composition operators on \?², J. Funct. Anal. 81 (1988), no. 2, 298 – 319. · Zbl 0669.47012 |

[10] | Carl C. Cowen and Barbara D. MacCluer, Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995. · Zbl 0873.47017 |

[11] | Carl C. Cowen and Barbara D. MacCluer, Some problems on composition operators, Studies on composition operators (Laramie, WY, 1996) Contemp. Math., vol. 213, Amer. Math. Soc., Providence, RI, 1998, pp. 17 – 25. · Zbl 0908.47025 |

[12] | Kevin W. Dennis, Co-hyponormality of composition operators on the Hardy space, Acta Sci. Math. (Szeged) 68 (2002), no. 1-2, 401 – 411. · Zbl 1063.47016 |

[13] | Peter L. Duren, Theory of \?^{\?} spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970. |

[14] | C. Hammond, On the norm of a composition operator with linear fractional symbol, Acta Sci. Math. (Szeged), to appear. · Zbl 1071.47508 |

[15] | Eric A. Nordgren, Composition operators, Canad. J. Math. 20 (1968), 442 – 449. · Zbl 0161.34703 |

[16] | D. B. Pokorny and J. E. Shapiro, Continuity of the norm of a composition operator, Integral Equations Operator Theory 45 (2003), 351-358. · Zbl 1053.47019 |

[17] | D. Q. Retsek, The Kernel Supremum Property and Norms of Composition Operators, Thesis, Washington University, 2001. |

[18] | W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987. · Zbl 0925.00005 |

[19] | H. Sadraoui, Hyponormality of Toeplitz and Composition Operators, Thesis, Purdue University, 1992. |

[20] | Joel H. Shapiro, The essential norm of a composition operator, Ann. of Math. (2) 125 (1987), no. 2, 375 – 404. · Zbl 0642.47027 |

[21] | Joel H. Shapiro, Composition operators and classical function theory, Universitext: Tracts in Mathematics, Springer-Verlag, New York, 1993. · Zbl 0791.30033 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.