ACTA issues

Reproducing kernels and norms of composition operators

Paul S. Bourdon, Dylan Q. Retsek

Acta Sci. Math. (Szeged) 67:1-2(2001), 387-394

Abstract. Motivated by a question raised by C. C. Cowen and B. D. MacCluer, we investigate when the norm of a composition operator $C_\phi $ on the classical Hardy space $H^2$ of the unit disk is determined by the action of $C_\phi $ or its adjoint $C_\phi ^*$ on the set $S$ of normalized reproducing kernels of $H^2$. Our results suggest that the action of $C_\phi ^*$ on $S$ rarely determines $\|C_\phi\|$. We show, for example, that when $C_\phi $ is compact then the action of $C_\phi ^*$ on $S$ determines the norm of $C_\phi $ if and only if $\phi(0) = 0$ or $\phi $ has the form $z\mapstochar\rightarrow sz + t$ for some constants $s$ and $t$ satisfying $|s| + |t| < 1$. We also show that when $\phi $ is inner and $\phi(0)\not=0$, then the action of $C_\phi ^*$ on $S$ determines $\|C_\phi\|$ if and only if $\phi $ is an automorphism of the unit disk.

AMS Subject Classification (1991): 47B33

Received August 30, 2000, and in revised form January 12, 2001. (Registered under 2790/2009.)