Noncommutative Varieties, Symmetric Weighted Fock Spaces, and Multipliers

摘要

In this paper we continue the study of noncommutative varieties VJ g-1 in polydomains in B(H)n1+center dot center dot center dot+nk associated with admissible k-tuples g := (g1,..., gk) of formal power series gi in indeterminates Zi,1,..., Zi, ni, and two-sided ideals J in the polynomial algebra generated by all the indeterminates. The main focus is on the admissible commutative varieties VJc g-1, where Jc is the two-sided ideal generated by the commutators Zi,j1Zs, j2 - Zs,j2Zi, ji. We introduce the symmetric weighted Fock space F 2 s (g) as a subspace of tensor products of full Fock spaces, which plays the role of model space for commutative varieties, and show that it can be identified with a reproducing kernel Hilbert space H2 (g) on a scalar polydomain in Cn1+center dot center dot center dot+nk, while the universal model {Li, j} of VJc g-1 is identified with the multipliers {M.i, j} by the coordinate functions on H2(g). We study the associated universal models for commutative varieties in connection with the Hardy algebras they generate and the dilation theory.