We prove an extension of the Wiener inversion theorem for convolution of summable series, allowing the terms to take values in a space of bounded linear operators. The resulting algebra is no longer commutative due to the composition of operators. Inversion theorems arise naturally in the context of proving dispersive estimates for the Schr\"odinger equation and lead to scale-invariant conditions for the class of admissible potentials. All results are joint work with Marius Beceanu.