Norm of the general polynomial differentiation composition operator from the space of Cauchy transforms to the mth weighted‐type space on the unit disk

• Published in: Mathematical methods in the applied sciences 2024-04, Vol.47 (6), p.3893-3902
• Main Author: Stević, Stevo
• Format: Article
• Language: English
• Subjects: Analytic functions; bounded operator; Composition; Differentiation; m$$ m $$th weighted‐type space; Mathematical analysis; operator norm; polynomial differentiation composition operator; Polynomials; space of Cauchy transforms
• ISSN: 0170-4214; 1099-1476
• DOI: 10.1002/mma.9681

We introduce the general polynomial differentiation composition operator Tφ→,ψ→nf(z)=∑j=0nψj(z)f(j)φj(z),$$ {T}_{\overrightarrow{\varphi},\overrightarrow{\psi}}^nf(z)=\sum \limits_{j=0}^n{\psi}_j(z){f}^{(j)}\left({\varphi}_j(z)\right), $$ where n∈N0$$ n\in {\mathrm{\mathbb{N}}}_0 $$, ψj$$ {\psi}_j $$, j=0,n ̅$$ j=\overline{0,n} $$, are holomorphic functions on the open unit disk D and φj$$ {\varphi}_j $$, j=0,n ̅$$ j=\overline{0,n} $$, are holomorphic self‐maps of D, calculate norm of the operator acting from the space of Cauchy transforms to the m$$ m $$th weighted‐type space, and characterize its boundedness, as well as the boundedness of the operator acting from the space of Cauchy transforms to the little m$$ m $$th weighted‐type space.