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.