On free resolutions in multivariable operator theory

Devin C. V. Greene

paper

On free resolutions in multivariable operator theory

arXiv:math/0109111

Abstract

We define the notion of a free resolution of a d-tuple $(T_{1}, T_{2}, ... T_{d})$ of mutually commuting operators acting on a Hilbert space H, and that this invariant gives rise to a class of vector space complexes parametrized by points in the unit ball $B_{d} = z \in C^{d} : ∣ z ∣ < 1$ . We show that for $λ \in B_{d}$ the homology of the corresponding complex is equivalent to the homology of the Koszul complex of a Mobius transform of $(T_{1}, T_{2}, ... T_{d})$ . The notion of a Mobius transform in multivariable operator is defined, and some of its properties are investigated.

23 pages