Non-selfadjoint Operator Algebras Generated by Weighted Shifts on Fock Space

Non-commutative multi-variable versions of weighted shifts arise naturally as ‘weighted’ left creation operators acting on Fock space. We investigate the unital wot-closed algebras they generate. The unweighted case yields non-commutative analytic Toeplitz algebras. The commutant can be described in terms of weighted right creation operators when the weights satisfy a condition specific to the non-commutative setting. We prove these algebras are reflexive when the eigenvalues for the adjoint algebra include an open set in complex n-space, and provide a new elementary proof of reflexivity for the unweighted case. We compute eigenvalues for the adjoint algebras in general, finding geometry not present in the single variable setting. Motivated by this work, we obtain general information on the spectral theory for non-commuting n-tuples of operators. The study of non-commutative multi-variable versions of weighted shift operators was initiated in [13]. These n-tuples arise naturally as ‘weighted’ left creation operators acting on Fock space. Certain C∗algebras determined by these weighted shifts on Fock space played a crucial role in [13], and the entire class is currently under investigation in [2]. In this paper, we consider non-selfadjoint algebras generated by these operators. In particular, we are interested in the weak operator topology closed non-selfadjoint algebras they generate. There is now an extensive body of literature for the unweighted case. The algebras generated by the left creation operators have been established as the appropriate non-commutative analytic Toeplitz algebras (for instance see [1, 5, 6, 14, 17, 18]). Our motivation with this work is twofold: we wish to establish nontrivial analogues of results obtained for standard weighted shifts. Towards this end we are motivated by the well-known survey article [21]. 2000 Mathematics Subject Classification. 47B37, 47L75, 46L54, 47A13. key words and phrases. Hilbert space, weighted shift, left creation operators, Fock space, commutant, reflexive algebra, joint spectral theory. 1 partially supported by a Canadian NSERC Post-doctoral Fellowship.