Summary: Let \(\mathbb{D}\) be the unit disc in the complex plane. Given a positive finite Borel measure \(\mu\) on the radius \([0, 1)\), we let \(\mu_n\) denote the \(n\)-th moment of \(\mu\) and we deal with the action on spaces of analytic functions in \(\mathbb{D}\) of the operator of Hibert-type \(\mathcal{H}_{\mu}\) and the operator of Cesàro-type \(\mathcal{C}_{\mu}\) which are defined as follows: If \(f\) is holomorphic in \(\mathbb{D}, f(z)=\sum_{n=0}^{\infty} a_n z^n (z\in\mathbb{D})\), then \(\mathcal{H}_{\mu} (f)\) is formally defined by \(\mathcal{H}_{\mu} (f)(z) = \sum_{n=0}^{\infty} \left( \sum_{k=0}^{\infty} \mu_{n+k}a_k\right) z^n (z\in\mathbb{D})\) and \(\mathcal{C}_{\mu} (f)\) is defined by \(\mathcal{C}_{\mu} (f)(z) = \sum_{n=0}^{\infty} \mu_n\left( \sum_{k=0}^n a_k\right) z^n (z\in\mathbb{D})\). These are natural generalizations of the classical Hilbert and Cesàro operators. A good amount of work has been devoted recently to study the action of these operators on distinct spaces of analytic functions in \(\mathbb{D}\). In this paper we study the action of the operators \(\mathcal{H}_{\mu}\) and \(\mathcal{C}_{\mu}\) on the Dirichlet space \(\mathcal{D}\) and, more generally, on the analytic Besov spaces \(B^p (1\leq p<\infty)\).