Generalized multiplication operators of higher order on weighted Hardy spaces

Abstract

In the present paper, the notion of generalized multiplication operators on weighted Hardy spaces is extended along with the description of its properties. The symbols inducing normal and compact generalized multiplication operators of higher order are identified. The commutativity of these operators with composition operators is also described.

On weighted generalized composition operators on weighted hardy spaces

“…We refer to book by Cowen and MacCluer [3] for basic theory of multiplication and composition operators. The study of these operators has been lifted from the Lebesgue spaces to other function spaces, for instance to Lorentz space, Orlicz space, Lorentz-Zygmund space, to name a few, over the years (see [1,4,10] and the references therein). Their theory touches various branches of mathematics and physics such as entropy theory, ergodic theory and classical mechanics.…”

“…The tendency of these operators to induce new classes of operators and their wide appearance have motivated mathematicians to come up with various generalizations of these operators. In the recent times, the generalized multiplication operators on weighted Hardy spaces have been introduced in [12] and further extended to k th -order generalized multiplication operators in [5], where k ≥ 1 is an integer. In a parallel manner, the study of generalized composition operators of higher order on weighted Hardy spaces H 2 (β) has been undertaken (see [4] and the references therein).…”

“…In the recent times, the generalized multiplication operators on weighted Hardy spaces have been introduced in [12] and further extended to k th -order generalized multiplication operators in [5], where k ≥ 1 is an integer. In a parallel manner, the study of generalized composition operators of higher order on weighted Hardy spaces H 2 (β) has been undertaken (see [4] and the references therein). Weighted composition operators on Hardy spaces are studied in [2] and [7].…”

“…Before we proceed ahead, it is imperative that we set up the necessary terminology required for the subsequent study. We recall from [4] that for a natural number k and an element f of H 2 (β) with expression…”

“…for each f ∈ H 2 (β) is continuous on H 2 (β), then this mapping is denoted by M φ,k and is called as k th -order generalized multiplication operator on H 2 (β) induced by φ (see [5]). Analogously, the continuous mapping on H 2 (β) defined as…”