Let \(H({\mathbf D})\) be the set of analytic functions in the unit disk \(\mathbf D\). A weight is a positive nonincreasing on \(|z|\) continuous function \(\nu (z)=\nu (|z|)\). The authors characterize the boundedness and compactness of weighted composition operators \(C_{\varphi, \psi}\) given by \(C_{\varphi, \psi}(f)=\psi(f\circ \varphi)\), where \(\varphi, \psi \in H({\mathbf D})\) and \(\varphi ({\mathbf D}) \subseteq {\mathbf D}\), acting between the spaces \(H^{\infty}_{\nu}= \{f \in H({\mathbf D}): \sup_{z \in {\mathbf D}} \nu(z) |f(z)|< \infty\}\) and \(H^0_{\nu}= \{f \in H({\mathbf D}): \lim_{|z|\rightarrow 1} \nu(z) |f(z)|=0\}\). They estimate the essential norm of the operator \(C_{\varphi, \psi}\) and compute it for the spaces \(H^0_{\nu}\) which are isomorphic to \(c_0\). It shown as well that being non compact a weighted composition operator is an isomorphism on a subspace isomorphic to \(c_0\) or \(\ell_{\infty}\). In the final section the authors study boundedness and compactness properties of composition operators acting between Bloch type spaces and little Bloch type spaces.