The Jumping Operator on Invariant Subspaces in Spaces of Analytic Functions

Abstract

Let D denote the Dirichlet space of holomorphic functions f in the open unit disc $$\mathbb {D}$$ with finite Dirichlet integral, $$\int _\mathbb {D}|f'|^2 dA < \infty $$ . For an $$M_z$$ -invariant subspace $$\mathcal {M}$$ of D we study the jumping operator $$P_\mathcal {M}M_z P_\mathcal {M}^{\perp }$$ from the orthogonal complement of $$\mathcal {M}$$ to $$\mathcal {M}$$ . We show that the jumping operator is in Schatten p-class for $$p > 1$$ and we obtain that for a zero-based invariant subspace $$\mathcal {M}$$ of D, the rank of the jumping operator is finite if and only if $$\mathcal {M}$$ 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.