On relative primeness of operator polynomials

Given operator polynomials A(λ) and B(λ), one of which is monic, conditions are given for the existence of operator polynomials C(λ) and D(λ) such that A(λ)C(λ) + B(λ)D(λ) = I for every $\text{λ ∈}\text{C}$. A special case will give a characterization of controlla- bility of an infinite-dimensional linear control system.