Let \(\mathscr {A}(\mathbb {U})\) denote the space of holomorphic functions on the unit disc \(\mathbb {U}\) , and let \(\mathscr {S}(\mathbb {U})\) be the family of analytic self-maps of \(\mathbb {U}\) . Fix \(n \in \mathbb {N}_0\) and consider analytic symbols \(\varvec{\Phi }=(\phi _j)_{j=0}^n \subset \mathscr {A}(\mathbb {U})\) together with a composition symbol \(\sigma \in \mathscr {S}(\mathbb {U})\) . This paper investigates the sum-type operator \( \mathcal {T}_{\varvec{\Phi },\sigma }^{\,n} h(u) = \sum _{j=0}^n \phi _j(u)\, h^{(j)}(\sigma (u)), \qquad h \in \mathscr {A}(\mathbb {U}), \) and derives necessary and sufficient conditions for its boundedness and compactness when acting from a Banach space \(\mathcal {X} \subset \mathscr {A}(\mathbb {U})\) into the weighted Zygmund-type spaces \(\mathfrak {Z}_{\omega }\) , \(\mathfrak {Z}_{\omega ,0}\) and the weighted Bloch-type spaces \(\mathfrak {B}_{\omega }\) , \(\mathfrak {B}_{\omega ,0}\) . These conditions are formulated in terms of weighted supremum estimates involving \(\varvec{\Phi }\) , \(\sigma \) , and their derivatives, under minimal structural assumptions on \(\mathcal {X}\) and the weight \(\omega \) . The results reveal how the boundary behavior of the inducing symbols determines the transition between boundedness and compactness.
