The Ces`aro-like operator on some analytic function spaces

Let $\mu$ be a finite positive Borel measure on the interval $[0,1)$ and $f(z)={\sum }_{n=0}^{\mathrm{\infty }}{a}_n{z}^n\in H(\mathbb{D})$. The Cesàro-like operator is defined by $${\mathcal{C}}_{\mu }(f)(z)=\sum _{n=0}^{\mathrm{\infty }}\left({\mu }_n\sum _{k=0}^n{a}_k\right)z^n,z\in \mathbb{D},$$ where, for $n\ge 0$, ${\mu }_n$ denotes the $n$-th moment of the measure $\mu$, that is, ${\mu }_n={\int }_{[0,1)}{t}^{n}d\mu (t)$. Let $X$ and $Y$ be subspaces of $H(\mathbb{D})$, the purpose of this paper is to study the action of ${\mathcal{C}}_{\mu }$ on distinct pairs $(X,Y)$. The spaces considered in this paper are Hardy space $H^p(0<p\le \mathrm{\infty })$, Morrey space $L^{2,\lambda }(0<\lambda \le 1)$, mean Lipschitz space, Bloch type space, etc.