Beurling-Lax representations using classical Lie groups with many applications. IV. GL(n,R), \(U^ *(2n)\), SL(n,\({\mathbb{C}})\), and a solvable group (Q1820377)

The Beurling-Lax theorem implies that any full range simply invariant subspace M of \(L^ 2_ n\) (square integrable vector valued functions on the unit circle) can be represented as \(\theta H^ 2_ n\) where \(\theta (e^{it})\) is unitary for almost all t. The authors here continue their study of characterizing subspaces M for which the Lie group U(n) of unitary matrices can be replaced by the other of the classical Lie groups. Applications include new types of symmetrized Wiener-Hopf factorization problems. The paper also includes a study of representations of certain types of non full range invariant subspaces; as an application there is obtained a geometric (module-theoretic) derivation of the Youla-Jabr-Bongiorno parametrization of all stabilizing compensators for a linear system. [For part III see Am. J. Math. 108, 95- 174 (1986; Zbl 0601.47015).]