Let \(B\) be the open unit ball in \(\mathbb{C}^N\) and \(\varphi: B \to B\) be a holomorphic self-map of \(B.\) \(C_{\varphi} = f \circ \varphi\) is a composition operator induced by \(\varphi\) where \(f\) is a holomorphic function on \(B.\)
The main result of this paper is the calculation of the essential norm of \(C_{\varphi}\) on the Bergman space \(A^2 (B)\) in the terms of the asymptotic upper bound of a quantity of the pull-back measure \(\mu\) induced by \(\varphi.\)