## Extremal non-compactness of composition operators with linear fractional symbol.(English)Zbl 1108.47024

The norm of certain composition operators with linear fractional symbol acting on the Hardy space in terms of the roots of associated hypergeometric functions is realized in this paper. The realization leads to simple necessary and sufficient conditions on $$\phi$$ for the composition operator to exhibit extremal non-compactness, establishes equivalence of cohyponormality and cosubnormality of composition operators with linear fractional symbol, and yields a complete classification of those linear fractional symbols that induce composition operators whose norms are determined by the action of the adjoint of the composition operator on the normalized reproducing kernels in Hardy space.

### MSC:

 47B33 Linear composition operators 47B38 Linear operators on function spaces (general)
Full Text:

### References:

 [1] Andrews, G.E.; Askey, R.; Roy, R., Special functions, (1999), Cambridge Univ. Press [2] Appel, M.; Bourdon, P.S.; Thrall, J., Norms of composition operators on the Hardy space, Experiment. math., 5, 111-117, (1996) · Zbl 0862.47015 [3] 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 [4] Bourdon, P.S.; Retsek, D.Q., Reproducing kernels and norms of composition operators, Acta. sci. math. (Szeged), 67, 555-562, (2001) · Zbl 1003.47017 [5] Cowen, C.C., Composition operators on spaces of analytic functions: A status report, (), 131-145 [6] Cowen, C.C., Linear fractional composition operators on $$H^2$$, Integral equations operator theory, 11, 151-160, (1988) · Zbl 0638.47027 [7] Cowen, C.C.; Kriete, T.L., Subnormality and composition operators on $$H^2$$, J. funct. anal., 81, 298-319, (1988) · Zbl 0669.47012 [8] Cowen, C.C.; MacCluer, B.D., Some problems in composition operators, (), 17-25 · Zbl 0908.47025 [9] Cowen, C.C.; MacCluer, B.D., Composition operators on spaces of analytic functions, (1995), CRC Press Boca Raton, FL · Zbl 0873.47017 [10] Dennis, K.W., Co-hyponormality of composition operators on the Hardy space, Acta. sci. math. (Szeged), 68, 401-411, (2002) · Zbl 1063.47016 [11] () [12] Hammond, C., On the norm of a composition operator with linear fractional symbol, Acta. sci. math. (Szeged), 69, 813-829, (2003) · Zbl 1071.47508 [13] Littlewood, J.E., On inequalities in the theory of functions, Proc. London math. soc., 23, 481-519, (1925) · JFM 51.0247.03 [14] Nordgren, E.A., Composition operators, Canad. J. math., 20, 442-449, (1968) · Zbl 0161.34703 [15] Retsek, D.Q., The kernel supremum property and norms of composition operators, (2001), thesis Washington University [16] Shapiro, J.H., The essential norm of a composition operator, Ann. of math., 125, 375-404, (1987) · Zbl 0642.47027 [17] Shapiro, J.H., What do inner functions know about composition operators?, Monatsh. math., 130, 57-70, (2000) · Zbl 0951.47026
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.