Norms of composition operators with rational symbol. (English) Zbl 1107.47018

Effinger-Dean, Sean; Johnson, Alan; Reed, Joseph; Shapiro, Jonathan

J. Math. Anal. Appl. 324, No. 2, 1062-1072 (2006).

Among the analytic selfmaps \(\phi\) on the unit disc which are non-inner rational functions in \({\mathbb C^*}\), the authors find a collection where the norm of the composition operator \(\| C_\phi\| \) on the Hardy space \(H^2\) can be computed from a given quadratic expression. They compare their results with those proved by C. Hammond [Acta Sci. Math. 69, No. 3–4, 813–829 (2003; Zbl 1071.47508)] in the case of linear fractional maps and the one by C. C. Cowen [Integral Equations Oper. Theory 11, No. 2, 151–160 (1988; Zbl 0638.47027)] for linear maps.

Reviewer: Oscar Blasco (Valencia)

MSC:

47B33	Linear composition operators
47B38	Linear operators on function spaces (general)
47A30	Norms (inequalities, more than one norm, etc.) of linear operators

Keywords:

composition operator; Hardy space; norm

Citations:

Zbl 1071.47508; Zbl 0638.47027

References:

[1]	Bourdon, P.S.; Fry, E.E.; Hammond, C.; Spofford, C.H., Norms of linear-fractional composition operators, Trans. amer. math. soc., 356, 2459-2480, (2004) · Zbl 1038.47500
[2]	Cowen, C., Linear fractional composition operators on \(H^2\), Integral equations operator theory, 11, 151-160, (1988) · Zbl 0638.47027
[3]	Cowen, C.; MacCluer, B., Composition operators on spaces of analytic functions, (1995), CRC Press Boca Raton, FL · Zbl 0873.47017
[4]	C. Hammond, On the norm of a composition operator, PhD thesis, University of Virginia, 2003 · Zbl 1071.47508
[5]	Hammond, C., On the norm of a composition operator with linear fractional symbol, Acta sci. math. (Szeged), 69, 813-829, (2003) · Zbl 1071.47508
[6]	Nordgren, E., Composition operators, Canad. J. math., 20, 442-449, (1968) · Zbl 0161.34703
[7]	Shapiro, J.H., Composition operators and classical function theory, (1993), Springer-Verlag New York · Zbl 0791.30033