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

Let $\mu$ be a finite positive Borel measure on the interval $[0, 1)$ and $f(z)=\sum_{n=0}^{\infty}a_{n}z^{n} \in H(\mathbb{D})$. The Ces\`aro-like operator is defined by $$\mathcal {C}_{\mu} (f)(z)=\sum^\infty_{n=0}\left(\mu_n\sum^n_{k=0}a_k\right)z^n, \ z\in \mathbb{D},$$ where, for $n\geq 0$, $\mu_n$ 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 $H^{p}(0<p\leq\infty)$, Morrey space $L^{2,\lambda}(0<\lambda\leq1)$, mean Lipschitz space, Bloch type space, etc.