Compact differences of composition operators from Bloch space to bounded holomorphic function space in the Polydisc

TL;DR

This paper provides estimates for the essential norm and characterizes the compactness of the difference between composition operators induced by holomorphic self-maps of the polydisc, acting from the Bloch space to bounded holomorphic functions.

Contribution

It offers new simple estimates for the essential norm and characterizes the compactness of differences of composition operators in the polydisc setting.

Findings

Derived simple estimates for the essential norm of operator differences.

Characterized when the difference of composition operators is compact.

Extended understanding of operator behavior in multivariable complex analysis.

Abstract

Let $\phi$ and $\psi$ be holomorphic self-maps of the unit polydisc $U^n$ in the $n$-dimensional complex space, and denote by $C_{\phi}$ and $C_{\psi}$ the induced composition operators. This paper gives some simple estimates of the essential norm for the difference of composition operators $C_{\phi}-C_{\psi}$ from Bloch space to bounded holomorphic function space in the unit polydisc. Moreover the compactness of the difference is also characterized.