The Hermite Property of a Causal Wiener Algebra Used in Control Theory

Let C+ := {s ∈ C | Re(s) ≥ 0} and let A denote the Banach algebra A = ( s(∈ C+) 7→ b fa(s) + ∞ X k=0 fke −stk ̨̨̨̨ fa ∈ L(0,∞), (fk)k≥0 ∈ `, 0 = t0 < t1 < t2 < . . . ) equipped with pointwise operations and the norm: ‖f‖ = ‖fa‖L1 + ‖(fk)k≥0‖`1 , f(s) = b fa(s) + ∞ X k=0 fke −stk (s ∈ C+). (Here b fa denotes the Laplace transform of fa.) It is shown that, endowed with the Gelfand topology, the maximal ideal space of A is contractible. In particular, the ring A is Hermite. The algebra A arises in control theory, and the Hermite property has useful consequences in the problem of stabilization of linear systems; see [3, Corollary 4.14]. The following statements are equivalent for f ∈ An×k, k < n: (1) There exists a g ∈ Ak×n such that gf = Ik on C+. (2) There exist F,G ∈ An×n such that GF = In on C+ and Fij = fij , 1 ≤ i ≤ n, 1 ≤ j ≤ k. (3) There exists a δ > 0 such that f(s)∗f(s) ≥ δIk, s ∈ C+.