The Jumping Operator on Invariant Subspaces in Spaces of Analytic Functions

Let D denote the Dirichlet space of holomorphic functions f in the open unit disc D with finite Dirichlet integral, integral(D) vertical bar f vertical bar(2)dA < infinity. For an M-z-invariant subspaceMof D we study the jumping operator PMMz P-M(perpendicular to) from the orthogonal complement of M to M. We show that the jumping operator is in Schatten p-class for p > 1 and we obtain that for a zero-based invariant subspaceMof D, the rank of the jumping operator is finite if and only ifMis 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.