Generalized Volterra-type integral operators between Bloch-type spaces

The Volterra-type integral operator plays an essential role in modern complex analysis and operator theory. Recently, Chalmoukis {Cn} introduced a generalized integral operator, say Ig,a, defined by Ig,af=Iⁿ(a₀f⁽ⁿ⁻¹⁾g'+a₁f⁽ⁿ⁻²⁾g''+⋯+aₙ₋₁fg⁽ⁿ⁾), where g∈ H(D) and a=(a₀,a₁,⋯,aₙ₋₁)∈ Cⁿ. Iⁿ is the nth iteration of the integral operator I. In this paper, we introduce a more generalized integral operators Ig⁽ⁿ⁾ that cover Ig,a on the Bloch-type space Bα, defined by Ig⁽ⁿ⁾f=Iⁿ(fg₀+⋯+f⁽ⁿ⁻¹⁾gₙ₋₁). We show the rigidity of the operator Ig⁽ⁿ⁾ and further the sum ∑ᵢ₌₁ⁿIgᵢNᵢ,kᵢ, where IgᵢNᵢ,kᵢf=INᵢ(f(kᵢ)gᵢ). Specifically, the boundedness and compactness of ∑ᵢ₌₁ⁿIgᵢNᵢ,kᵢ are equal to those of each IgᵢNᵢ,kᵢ. Moreover, the boundedness and compactness of Iⁿ((fg')⁽ⁿ⁻¹⁾) are independent of n when α>1.