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

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.

### 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
Full Text:

### 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
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