Invertible Factorization over Multiplier Algebras

Let \({\mathcal{A}}\) denote the multiplier algebra of an E-valued reproducing kernel Hilbert space, \({H_E^2(k)}\) . Then when H 2(k) is nice, we give necessary and sufficient conditions that T > 0 factors as A*A, where A and \({A^{-1} \in \mathcal{A}}\) . Such nice spaces include the Bergman and Hardy spaces on the unit polydisk and unit ball in \({\mathbb{C}^d}\) .