Sum of generalized weighted composition operators between weighted spaces of analytic functions

Let H(D) be the space of analytic functions on the unit disc and let (D) denote the set of analytic self-maps of D. Let Ψ = (ψj)j=0k be such that ψj ∈ H(D) and φ ∈ S(D). We characterize the boundedness, compactness and completely continuous of the sum of generalized weighted composition operators 'Equation Presented' between weighted Banach spaces of analytic functions Hv∞(Hv0) and Hw∞(Hw0) which unifies the study of products of composition operators, multiplication operators and differentiation operators. As applications, we obtain the boundedness and compactness of the generalized weighted composition operators Dψ,φn: v(ßv0) → w(ßw0), v(ßv0) → Hw∞(Hw0) and Hv∞(Hv0) → w(ßw0), where v(ßv0) and w(ßw0) are weighted Bloch-type (little Bloch-type) spaces. Also, new characterizations of the boundedness and compactness of the operators TΨ,φk and DΨ,φn are given. Examples of bounded, unbounded, compact and non-compact operators TΨ,φk and DΨ,φn are given to explain the role of inducing functions Ψj, φ and the weights v, w of the underlying weighted spaces.