Difference of composition operators on weighted Bergman spaces over the half-plane

Abstract

Recently, the bounded, compact and Hilbert-Schmidt difference of composition operators on the Bergman spaces over the half-plane are characterized in (Choe et al. in Trans. Am. Math. Soc., 2016, in press). Motivated by this, we give a sufficient condition when two composition operators $C_{\varphi}$ and $C_{\psi}$ are in the same path component under the operator norm topology and show that there is no cancellation property for the compactness of double difference of composition operators. More precisely, we show that if $C_{\varphi_{1}}$ , $C_{\varphi_{2}}$ , and $C_{\varphi_{3}}$ are distinct and bounded, then $(C_{\varphi _{1}}-C_{\varphi_{2}})-(C_{\varphi_{3}}-C_{\varphi_{1}})$ is compact if and only if both $C_{\varphi_{1}}-C_{\varphi_{2}}$ and $C_{\varphi _{1}}-C_{\varphi_{3}}$ are compact on weighted Bergman spaces over the half-plane. Moreover, we prove the strong continuity of composition operators semigroup induced by a one-parameter semigroup of holomorphic self-maps of half-plane.