  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.
