The Schur algorithm for generalized Schur functions. III: \(J\)-unitary matrix polynomials on the circle. (Q1399923)

[Parts I, II of this series of four papers were reviewed in Zbl 1008.47017 and Zbl 1043.47014; for part IV, see Zbl 1081.47013.] For each non-negative integer \(\kappa\), define the set \(S_\kappa\) to consist of all those complex-valued functions \(s\) that are meromorphic on the unit disc \({\mathbb D}\), have \(\kappa\) poles in the disc, and satisfy \(\limsup_{r\uparrow 1} {| s(re^{it})| }\leq 1\) for almost all \(t\in [0,\,2\pi]\). Equivalently, \(S_\kappa\) consists of all functions \(s\), meromorphic in the disc, for which the kernel \[ S_s(z,\,w) := {{1-s(z)s(w)^*}\over{1-zw^*}} \] has \(\kappa\) negative squares. The union \(S:=\bigcup_{\kappa =0}^{\infty}{S_\kappa}\) is the set of generalized Schur functions. (The classical Schur functions comprise \(S_0\).) The set of those elements of \(S_\kappa\) that are analytic in some neighborhood of the origin will be denoted \({S_\kappa}^0\). The classical Schur algorithm, applied to a function \(s_0 \in S_0\), yields a sequence of functions \(\{s_j\}\), each in \(S_0\), defined recursively by \[ s_{j+1}(z) = {{1}\over{z}}{{s_j(z)-s_j(0)}\over{1-s_j(z)s_j(0)^*}} \] for \(j\geq 0\). The generalized Schur algorithm applies to all functions in the set \(S\) not of constant modulus one. The recursion is the same as in the classical case provided that \(| s_j(0)| <1\), but variations on this pattern can also arise when \(| s_j(0)| \geq 1\). In particular, the case \(| s_j(0)| =1\) tends to be rather complicated. Here, the authors introduce an alternative point of view by replacing the function \(s\) in \(S\) with a homogeneous pair \((u,\,v)\), where \(u\) and \(v\) are analytic in the open unit disc, have no common zeros (so \(| u(0)| +| v(0)| \neq 0\)), and satisfy \(s(z)=v(z)/u(z)\). In this context, the Schur kernel is \[ S_{(u,\,v)}(z,w) := {{u(z)u(w)^*-v(z)v(w)^*}\over{1-zw^*}}, \] for \(z\) and \(w\) in the disc. For a function \(s\in S_\kappa\), the Schur kernel, \(S_s\) defined above, generates a reproducing kernel Pontryagin space, denoted \({\mathcal P}(s)\). (For \(\kappa=0\), this is a reproducing kernel Hilbert space.) The space \({\mathcal P}(s)\) is isomorphic to the space \({\mathcal P}(u,v)\) generated by the kernel \(S_{(u,\,v)}\). In the step-by-step implementation of the generalized Schur algorithm in the homogeneous point of view, a key role is played by the set \({\mathcal U}_J\) of \(J\)-unitary matrix-valued polynomials. Specifically, with \(J:=\begin{pmatrix}1&0\cr 0&-1\end{pmatrix}\), a \(J\)-unitary polynomial \(\Theta\) is a \(2\times 2\) complex matrix-valued polynomial on the unit circle that satisfies \[ \Theta(z)J\Theta(z)^*=J \] for all \(z\) on the circle. For \(\Theta\in{\mathcal U}_J\), set \({\mathcal P}(\Theta) := H_{2,J}\ominus \Theta H_{2,J}\), where \(H_{2,J}\) is the function space \(H_2\oplus H_2\) equipped with the inner product \(\langle f,\,g\rangle_J := \langle f,\,Jg\rangle_2\). Then (see Theorem 4.2) \({\mathcal P}(\Theta)\) is a reproducing kernel Pontryagin space with kernel \[ K_\Theta(z,w)={{J-\Theta(z)J\Theta(w)^*}\over{1-zw^*}}, \] for \(z\) and \(w\) in the disc. The elements of \({\mathcal P}(\Theta)\) are \({\mathbb C}^2\)-valued polynomials and \({\mathcal P}(\Theta)\) is invariant under the backward-shift operators \(R_a\) defined by \(R_a f(z)=(f(z)- f(a))/(z-a)\). Moreover, any finite-dimensional non-degenerate \(R_0\)-invariant subspace of \(H_{2,J}\) whose elements are polynomials is of the form \({\mathcal P}(\Theta)\) for some \(\Theta\in {\mathcal U}_j\). In section 5, the authors show (see Proposition 5.1 and Theorems 5.3 and 5.4) that every \(\Theta\in {\mathcal U}_J\) can be uniquely factored in the form \[ \Theta(z) = z^n \theta_1(z)\cdots \theta_N(z)U, \] where \(n\) is a natural number, \(U=\Theta(1)\) is a constant, and the \(\theta_j\) are normalized elementary factors, of which there are exactly two distinct types. The connection between the Pontryagin spaces \({\mathcal P}(u,v)\) and \({\mathcal P}(\Theta)\), and thus the role of \(J\)-unitary polynomials in the Schur algorithm, is revealed in section 6. For a given homogeneous pair \((u,v)\), the authors identify a nested chain \(\{M_N\}\) of subspaces of \({\mathcal P}(u,v)\) such that the union \(M=\bigcup{M_n}\) is dense in \({\mathcal P}(u,v)\). Proposition 6.2 shows that, if the subspace \(M_n\) is non-degenerate in the \(H_{2,J}\) inner product, then there exists \(\Theta_N\in{\mathcal U}_J\) such that \(M_N={\mathcal P}(\Theta_N)\). Moreover, if \(N\) is the first index for which \(M_N\) is non-degenerate, then the corresponding \(\Theta_N\) is an elementary factor. Proposition 6.4 and Theorems 6.5 and 6.6 then imply the step-by-step Schur algorithm for homogeneous pairs. If \((u_0,v_0):=(u,v)\) is a homogeneous pair and, working recursively, if \(\theta_j\in{\mathcal U}_J\) is the first elementary \(J\)-unitary polynomial for the pair \((u_{j-1},\,v_{j-1})\) (for \(j=1,\,2,\ldots\)), then \[ (u_{j-1}(z)-v_{j-1}(z))\theta_j(z)=z^{\text{ deg}\theta_j}(u_{j}(z)-v_{j}(z)). \] Letting \(\Theta_j(z)=\theta_1(z)\cdots\theta_j(z)\) for each \(j\), it follows that \((u(z)-v(z))\Theta_j(z)=z^{\text{ deg}\Theta_j}(u_{j}(z)-v_{j}(z))\). The paper concludes with some examples in section 7.