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

Abstract

Noncommutative multivariable versions of weighted shifts arise naturally as `weighted' left creation operators acting on Fock space. We investigate the unital weak operator topology closed algebras they generate. The unweighted case yields noncommutative analytic Toeplitz algebras. The commutant can be described in terms of weighted right creation operators when the weights satisfy a condition specific to the noncommutative 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 noncommuting $n$-tuples of operators.