Beurling-Lax representations using classical Lie groups with many applications. II: \(\mathrm{GL}(n,\mathbb C)\) and Wiener-Hopf factorization
The classical Beurling-Lax-Helson-Halmos theorem characterizes the full range simply invariant subspaces \(M\) of \(L^ 2_ n\) as being of the form \(M=\Theta H^ 2_ n\) where \(\Theta \in L^{\infty}_{n\times n}\) is a phase function. Here \(L^ 2_ n\) is the Hilbert space of measurable \(\mathbb C^ n\)-valued functions on the unit circle \(\{e^{it}: 0\leq t\leq 2\pi \}\) which are square integrable in norm, \(H^ 2_ n\) is the Hardy subspace of functions in \(L^ 2_ n\) with vanishing negative Fourier coefficients, \(L^{\infty}_{n\times n}\) consists of the measurable bounded \(n\times n\)-matrix functions on the unit circle, and a phase function is one whose values \(\Theta(e^{it})\) are unitary for a.e. \(t\). From this result there follows the classical inner--outer factorization for a large class of analytic matrix functions. The well-known Halmos proof of the above theorem represents \(M\) as an orthogonal direct sum \[ M=\oplus^{\infty}_{j=0}\chi^ j{\mathcal L}\qquad(\text{where } \chi(e^{it})=e^{it}) \] where \({\mathcal L}\) is the ``wandering subspace'' \({\mathcal L}=M\theta \chi M.\) This paper presents a characterization (roughly) of pairs of subspaces \((M^ x,M)\) contained in \(L^ 2_ n\) such that simultaneously \(M=\Theta H^ 2_ n\) and \(M^ x=\Theta H^{2\perp}\) for some matrix function \(\Theta\) whose values \(\Theta(e^{it})\) are not necessarily unitary. This representation theorem turns out to be equivalent to Wiener-Hopf factorization \((F=F_-F_+\) where \(F_-\) is analytic and invertible outside the disk, \(F_+\) is analytic and invertible inside the disk) in much the same way that the original Beurling theorem is equivalent to inner--outer factorization. The proof involves analyzing the non-orthogonal wandering subspace decompositions \[ M=+^{\infty}_{j=0}\chi^ j{\mathcal L},M^ x=\dot +^{\infty}_{j=- 1}\chi^ j{\mathcal L}* \] and \[ L^ 2_ n=+^{\infty}_{j=-\infty}\chi c^ j{\mathcal L} \] where \({\mathcal L}=\chi M^ x\cap M\). Also presented is a more general representation theorem corresponding to noncanonical Wiener-Hopf factorization; with \(F_-,F_+\) not invertible throughout the disk. [For part I see J. Oper. Theory 9, 107--142 (1983; Zbl 0505.47029)].
scientific article; zbMATH DE number 3857784