Let D denote the Dirichlet space of holomorphic functions f in the open unit disc D\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {D}$$\end{document} with finite Dirichlet integral, ∫D|f′|2dA<∞\documentclass[12pt]{minimal}
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\begin{document}$$\int _\mathbb {D}|f'|^2 dA < \infty $$\end{document}. For an Mz\documentclass[12pt]{minimal}
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\begin{document}$$M_z$$\end{document}-invariant subspace M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}$$\end{document} of D we study the jumping operator PMMzPM⊥\documentclass[12pt]{minimal}
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\begin{document}$$P_\mathcal {M}M_z P_\mathcal {M}^{\perp }$$\end{document} from the orthogonal complement of M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}$$\end{document} to M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}$$\end{document}. We show that the jumping operator is in Schatten p-class for p>1\documentclass[12pt]{minimal}
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\begin{document}$$p > 1$$\end{document} and we obtain that for a zero-based invariant subspace M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}$$\end{document} of D, the rank of the jumping operator is finite if and only if M\documentclass[12pt]{minimal}
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\begin{document}$$\mathcal {M}$$\end{document} is of finite codimension. We also prove that there are invariant subspaces of D which have infinite codimension such that the corresponding jumping operators have finite rank. Furthermore, we show that some similar results hold in the setting of the Bergman space.
