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

Let be the space of analytic functions on the unit disc and let denote the set of analytic self-maps of . Let ψ = (ψj)j=0k be such that ψj and φ . We characterize the boundedness, compactness and completely continuous of the sum of generalized weighted composition operators Tψ,φkf =Ζ j=0kψ j ⋅ f(j) φ =Ζ j=0k ψj,φjf,f , between weighted Banach spaces of analytic functions Hv∞(H v0) and Hw∞(H w0) 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 ψ,φn:Ζ v(Ζv0) →Ζ w(Ζw0), Ζv(Ζv0) → H w∞(H w0) and Hv∞(H v0) →Ζ w(Ζw0), where Ζv(Ζv0) and Ζw(Ζw0) are weighted Bloch-type (little Bloch-type) spaces. Also, new characterizations of the boundedness and compactness of the operators Tψ,φk and ψ,φn are given. Examples of bounded, unbounded, compact and non-compact operators Tψ,φk and ψ,φn are given to explain the role of inducing functions ψj, φ and the weights v, w of the underlying weighted spaces.