Triplets of Closely Embedded Dirichlet Type Spaces on the Unit Polydisc

Abstract

We propose a general concept of triplet of Hilbert spaces with closed embeddings, instead of continuous ones, and we show how rather general weighted L2 spaces yield this kind of generalized triplets of Hilbert spaces for which the underlying spaces and operators can be explicitly calculated. Then we show that generalized triplets of Hilbert spaces with closed embeddings can be naturally associated to any pair of Dirichlet type spaces Dα(DN) of holomorphic functions on the unit polydisc DN and we explicitly calculate the associated operators in terms of reproducing kernels and radial derivative operators. We also point out a rigging of the Hardy space H2(DN) through a scale of Dirichlet type spaces and Bergman type spaces. © 2012 Springer Basel.