Extended Cesàro operators on the Bloch space in the unit ball of \(\mathbb{C}^n\). (English) Zbl 1044.47023

Hu, Zhangjian

Acta Math. Sci., Ser. B, Engl. Ed. 23, No. 4, 561-566 (2003).

Summary: The paper defines an extended Cesàro operator \(T_g\) with holomorphic symbol \(g\) in the unit ball \({\mathbf B}\) of \(\mathbb{C}^n\) as \[ T_g(f)(z)= \int^1_0 f(tz)\text{Re\,}g(tz){dt\over t},\quad f\in H({\mathbf B}),\;z\in{\mathbf B}, \] where \(\text{Re\,}g(z)= \sum^n_{j=1} z_j{\partial g\over\partial z_j}\) is the radial derivative of \(g\). The author characterizes those \(g\) for which \(T_g\) is bounded (or compact) on the Bloch space \({\mathcal B}\) and the little Bloch space \({\mathcal B}_0\).

MSC:

47B38	Linear operators on function spaces (general)
32A36	Bergman spaces of functions in several complex variables

Keywords:

boundedness; extended Cesàro operator; Bloch space