ACTA issues

Norm calculations of composition operators on Fock spaces

Ovidiu Furdui

Acta Sci. Math. (Szeged) 74:1-2(2008), 281-288

Abstract. The paper deals with the norm calculations of the composition operator on Fock space over ${\msbm C}$. If $0< p< \infty $ and $C_{\varphi }\colon F^p\rightarrow F^p$ is the composition operator defined by $C_{\varphi }f=f\circ\varphi $, then it has been shown that the composition operator is bounded if and only if $\varphi(z)=az+b$, where either $|a|< 1$ and $b\in{\msbm C}$ or $|a|=1$ and $b=0$. Further, when $p=2$, it was proved that $\|C_{\varphi }\|_{2}=e^{{|b|^2\over4(1-|a|^2)}}$. In this paper we prove that for $p>2$ and $|a|< 1$ the norm of the composition operator is $\|C_{\varphi }\|_{p}=e^{|b|^2\over2p(1-|a|^2)}$.

AMS Subject Classification (1991): 47B33

Keyword(s): Composition operator, Fock space

Received March 1, 2007, and in revised form September 8, 2007. (Registered under 6018/2009.)