In this paper, we study a problem of the boundedness of integral operators on Hardy–Sobolev space. By using an equivalence between this kind of space and weighted Bergman space, we transform the corresponding problem to the latter. That is, we show that for an analytic function \(\varphi \) on the unit ball \(\mathbb {B}_n\) and \(\beta <0\) , the integral operator \(\begin{aligned} S_\varphi f(z)=\int \limits _{\mathbb {B}_n}f(w)\varphi (z\cdot {\overline{w}})dv_{-1-2\beta }(w), \qquad z\in \mathbb {B}_n \end{aligned}\) is bounded on \(A^2_{-1-2\beta }(\mathbb {B}_n)\) if and only if there exists a sequence \(\{a_k\}\in l^\infty (\mathbb {Z}_+^n)\) such that \(\begin{aligned} \varphi (z)=\sum _{k\in \mathbb {Z}_+^n}\frac{\Gamma (n+|k|-2\beta )}{k!\Gamma (n-2\beta )}a_k z^k, \end{aligned}\) where \(\cdot \) represents coordinatewise multiplication, \(\mathbb {Z}_+^n\) is the set of multi-indexes of nonnegative integers and \(dv_{-1-2\beta }(w)=c_{-1-2\beta }(1-|z|^2)^{-1-2\beta }dv(w)\) is the normalized weighted volume measure. Then we indicate that the set of all bounded operators \(S_\varphi \) happen to be the set of the radial operators on \(A^2_{-1-2\beta }(\mathbb {B}_n)\) . Besides, we get the adjoint operator, normality, compactness and spectrum of the bounded operator \(S_\varphi \) . Moreover, the \(C^*-\) algebra property of the Banach space \(\mathcal {L}(A^2_{-1-2\beta })\) , consisting of bounded linear operators on \(A^2_{-1-2\beta }(\mathbb {B}_n)\) , and even the common reducing subspaces of all bounded \(S_\varphi \) , are both characterized.
