Beurling-Lax representations using classical Lie groups with many applications II: GL(n,©) and Wiener-Hopf factorization

Abstract

The classical Beurling-Lax-invariant subspace theorem characterizes the full range simply invariant subspacesM of L ²_n as being of the formM=ΘH ²_n where Θ∈L ^∞_n×n is a phase function. Here L ²_n is the Hilbert space of measurable ℂⁿ-valued functions on the unit circle {e^it|0≤t≤2π} which are square-integrable in norm, H ²_n is the subspace of functions in L ²_n with analytic continuation to the interior of the disk {z‖z|<1}, L ²_n×n is the space of measurable essentially bounded n×n matrix functions on the unit circle, and a phase function Θ is one whose values Θ(e^it) are unitary for a.e. t (i.e., Θ(e^it) is in the Lie group U(n) a.e.). Halmos extended this to L ²_∞ . A subspace M⊂L ²_n is said to beinvariant if e^it M⊂M,simply invariant if in addition$\mathop \cap \limits_{k \geqslant 0}$ e^ikt M=(0), andfull range if$\mathop \cup \limits_{N > 0}$ e^−iNt M is dense in L ²_n . In the Beurling-Lax representationM=ΘH ²_n ,M uniquely determines Θ up to a unitary constant factor on the right if one insists that Θ(e^it)∈U(n). If one demands only that Θ(e^it) ∈ GL(n,ℂ) (the group of invertible n×n complex matrices), however, there is considerably more freedom; in fact ΘH ²_n =Θ₁H ²_n where Θ₁ ΘF and F∈L ^∞_n×n is outer with inverse F⁻¹∈L ^∞_n×n . More generally, we have ΘH ²_n =[Θ₁H ^∞_n ]⁻ whenever Θ₁=ΘF and F is outer with F and F⁻¹ in L ²_n×n . (An F∈L ²_n×n will be said to beouter if FH ^∞_n is a dense subset of H ²_n .) In particular one can use this freedom to obtain representationsM=[ΘH ^∞_n ]⁻ where the representor Θ has values Θ(e^it) in other matrix Lie groups. This program was carried out in accompanying work of the authors [B-H1-4] for the classical simple Lie groups U(m,n), O(p,q), O^*(2n), Sp(n,C), Sp(n,R), Sp(p,q), O(n,C), GL(n,R), U^*(2n), GL(n,R) and SL(n,C) and many applications were given. In this paper we give a natural theorem for GL(n,ℂ), by introducing the extra structure of preassigning the spaceM ^x=[ΘH ^∞_n ]⁻ as well asM=[ΘH ^∞_n ]⁻. The theorems in [B-H1-4] can be derived by specializing our main result here for GL(n,ℂ) to the various subgroups which we listed.