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

Abstract

Let [Formula: see text] be the space of analytic functions on the unit disc [Formula: see text] and let [Formula: see text] denote the set of analytic self-maps of [Formula: see text]. Let [Formula: see text] be such that [Formula: see text] and [Formula: see text]. We characterize the boundedness, compactness and completely continuous of the sum of generalized weighted composition operators [Formula: see text] between weighted Banach spaces of analytic functions [Formula: see text] and [Formula: see text] 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 [Formula: see text], [Formula: see text] and [Formula: see text], where [Formula: see text] and [Formula: see text] are weighted Bloch-type (little Bloch-type) spaces. Also, new characterizations of the boundedness and compactness of the operators [Formula: see text] and [Formula: see text] are given. Examples of bounded, unbounded, compact and non-compact operators [Formula: see text] and [Formula: see text] are given to explain the role of inducing functions [Formula: see text], [Formula: see text] and the weights [Formula: see text], [Formula: see text] of the underlying weighted spaces.