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 ${\mathbb{C}^d}$ .