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

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
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