Polynomials with Given Spectral Pairs and Exactly Controllable Systems

Abstract

In this chapter we consider the following problem: Given a pair of operators X ∈ L(y, x), T ∈ L(y) (here x and y are Banach spaces), construct, if possible, an operator polynomial L(λ) whose right spectral pair (with respect to the whole complex plane) is (X, T). By Theorem 6.6.1, a necessary condition is that col[XTⁱ] ^p−1_i=0 is left invertible for some p; we shall see that this condition (if x, y are Hilbert spaces) is also sufficient. It turns out that this problem is very closely related to the problems concerning spectrum assignment in exactly controllable systems, an important topic in the modern systems theory.