Products of Composition, Multiplication and Iterated Differentiation Operators Between Banach Spaces of Holomorphic Functions

Abstract

Let H(D) denote the space of holomorphic functions on the unit disk D of C, ψ, ϕ ∈ H(D), ϕ(D) ⊂ D and n ∈ N ∪ {0}. Let C ϕ , M ψ and D n denote the composition, multiplication and iterated differentiation operators, respectively. To treat the operators induced by products of these operators in a unified manner, we introduce a sum operator n j=0 M ψj C ϕ D j . We characterize the boundedness and compactness of this sum operator mapping from a large class of Banach spaces of holomorphic functions into the kth weighted-type space W, and give its estimates of norm and essential norm. Our results show that the boundedness and compactness of the sum operator depend only on the symbols and the norm of the point-evaluation functionals on the domain space. Our results cover many known results in the literature. Moreover, we introduce the order boundedness of the sum operator and turn its study into that of the boundedness and compactness.

On a Sum of More Complex Product-Type Operators from Bloch-Type Spaces to the Weighted-Type Spaces

“…There may be many people who have the same idea as us. Actually, the authors in [18] introduced the following operator, which achieved the expectations…”

Polynomial differentiation composition operators from Besov‐type spaces into Bloch‐type spaces

“…where n, k ∈ N 0 , 𝜑 ∈ S(D), and 𝜓 𝑗 ∈ H(D), 𝑗 = 0, 1, … , k, as well as some related operators on C n (for some results in the direction, see previous studies [40][41][42][43]). The case n = 0 has been recently studied in Hu et al [44] and Wang et al [45]. A special case was also studied in Sharma and Sharma [46].…”

“…Having published [35], Stević proposed to his collaborators investigation of the operator where $$$ n,k\in {\mathrm{\mathbb{N}}}_0 $$$, , and , as well as some related operators on $$$ {\mathrm{\mathbb{C}}}^n $$$ (for some results in the direction, see previous studies [40–43]). The case $$$ n=0 $$$ has been recently studied in Hu et al [44] and Wang et al [45]. A special case was also studied in Sharma and Sharma [46].…”

Stević–Sharma‐type operators between Bergman spaces induced by doubling weights

“…In the following years, many authors studied Stevi ć-Sharma-type operators. See, for example, [26][27][28][29][30][31][32] for more investigations about these operators on the spaces of holomorphic functions in the unit disc. Stevi ć and Sharma also proposed studying their operators between spaces of holomorphic functions on the upper half-plane [33].…”