Summary: Let \(A\) be a positive bounded operator on an infinite dimensional complex and separable Hilbert space \((\mathcal{H}, \langle, \rangle)\) and \(\mathfrak{A} : = \mathcal{B}(\mathcal{H})/\mathcal{K}(\mathcal{H})\) be the Calkin algebra and \(\mathfrak{A}'\) its dual; here \(\mathcal{K}(\mathcal{H})\) denotes the closed ideal of \(\mathcal{B}(\mathcal{H})\) consisting of all compact operators on \(\mathcal{H}\). For an operator \(T\) on \(\mathcal{H}\), let \(\Vert T\Vert_A\) and \(V_A(T)\) denote the \(A\)-operator semi-norm and the \(A\)-numerical range induced by \(A\). The \(A\)-essential numerical range and the \(A\)-essential norm of \(T\) are defined by \(V_A^e(T)=V_{\widehat{A}}(\widehat{T})\) and \[\begin{aligned} \Vert T\Vert_A^e{:}{=}\sup_{{\mathcal{S}}_{\widehat{A}}({\mathfrak{A}})}\sqrt{f\left( \widehat{T^*} \widehat{A}\widehat{T}\right)}, \end{aligned} \] where \(\mathcal{S}_{\widehat{A}}({\mathfrak{A}})=\left\{f\in{\mathfrak{A}}': f\geq 0, f(\widehat{A})=1\right\}\) and \(\widehat{T}{:}{=}T+{\mathcal{K}}(\mathcal{H})\) is the coset containing \(T\). In this paper, we establish some properties of the \(A\)-essential numerical range. In particular we show that if \(T\) has an \(A\)-adjoint then \[ \begin{aligned} V_A^e(T)=\bigcap_{K\in \mathcal{K}_A(\mathcal{H})} V_A(T+K). \end{aligned} \] Further, we prove that \(\lambda \in V_A^e(T)\) if and only if there exists a sequence \((h_n)\in \mathcal{H}\) of \(A\)-unit vectors such that \(h_n\rightarrow 0\) weakly and \(\left\langle ATh_n, h_n\right\rangle \rightarrow \lambda\). Other auxiliary results are established.