Coprime factorizations and stabilizability of infinite-dimensional linear systems

It has been known that a matrix-valued transfer function is dynamically stabilizable iff it has a doubly coprime factorization. We extend this to operator-valued functions and also to controllers with internal loop. We then present several other equivalent conditions, such as having a stabilizable and detectable realization. Our results lead to the extension of the classical results on dynamic stabilization and dynamic partial stabilization to general proper operator-valued functions. Part of the results are new even for scalar-valued functions.