Aspects of multivariable operator theory on weighted symmetric Fock spaces

Abstract

We obtain all Dirichlet spaces ℱq, q ∈ ℝ, of holomorphic functions on the unit ball of ℂN as weighted symmetric Fock spaces over ℂN. We develop the basics of operator theory on these spaces related to shift operators. We do a complete analysis of the effect of q ∈ ℝ in the topics we touch upon. Our approach is concrete and explicit. We use more function theory and reduce many proofs to checking results on diagonal operators on the ℱq. We pick out the analytic Hilbert modules from among the ℱq. We obtain von Neumann inequalities for row contractions on a Hilbert space with respect to each ℱq. We determine the commutants and investigate the almost normality of the shift operators. We prove that the C*-algebras generated by the shift operators on the ℱq fit in exact sequences that are in the same Ext class. We identify the groups K0 and K1 of the Toeplitz algebras on the ℱq arising in K-theory. Radial differential operators are prominent throughout. Some of our results, especially those pertaining to lower negative values of q, are new even for N = 1. Many of our results are valid in the more general weighted symmetric Fock spaces ℱb that depend on a weight sequence b.