Product-type operators on weak vector valued $\alpha$-Besov spaces

Authors: SEPIDEH NASRESFAHANI, EBRAHIM ABBASI

Abstract: Let $\psi_1$ and $\psi_2$ be analytic functions on the open unit disk $\mathbb{D}$ and $\phi$ an analytic self map on $\mathbb{D}$. Let $M_\psi$, $C_\phi$ and $D$ denote the multiplication, composition and differentiation operators. We consider operators $M_{\psi_1} C_\phi$, $M_{\psi_2} C_\phi D$ and the Stevi\'c-Sharma operator $T_{\psi_1,\psi_2,\phi}(f)=M_{\psi_1}C_\phi (f)+M_{\psi_2}C_\phi D(f)$ on $\alpha$-Besov space $\mathcal{B}_{p,\alpha}$ and weak vector valued $\alpha$-Besov space $ w\mathcal{B}_{p,\alpha}(X)$ for complex Banach space $X$ and find some equivalent statements for boundedness of these operators. Also, boundedness and compactness of composition operator $C_\phi$ on $\mathcal{B}_{p,\alpha}(\mathbb{D})$ and $w\mathcal{B}_{p,\alpha}(\mathbb{D})$ are given.

Keywords: Product of composition multiplication and differentiation, $\alpha$-Besov spaces, Carleson measure, weak vector valued $\alpha$-Besov spaces, boundedness

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