Criteria for the strong regularity of \(J\)-inner functions and \(\gamma\)-generating matrices.
Let \(J\) be an \(m\times m\) signature matrix \((J= J^*= J^{-1})\), and let \(\Omega_+\) be either the open unit disk or the open upper halfplane. An \(m\times m\)-matrix valued function \(U(\lambda)\) that is meromorphic in \(\Omega_+\) is called \(J\)-inner if \(U(\lambda)\) is \(J\)-unitary for a.e. point \(\lambda\) on the boundary of \(\Omega_+\) and \(J- J(\lambda)^* JU(\lambda)\) is positive semidefinite for every point \(\lambda\in \Omega_+\) of analyticity of \(U(\lambda)\). The class of right strongly regular \(J\)-inner functions was introduced in previous works of the authors in connection with their study of canonical integral and differential systems. For \(J= \left[ \begin{smallmatrix} I_p &0\\ 0 &-I_q \end{smallmatrix} \right]\) (here \(p+q= m\)), a \(J\)-inner function \(U(\lambda)= \left[ \begin{smallmatrix} w_{11}(\lambda) &w_{12}(\lambda)\\ w_{21}(\lambda) &w_{22}(\lambda) \end{smallmatrix} \right]\), where \(w_{11}(\lambda)\) is \(p\times p\) and \(w_{22}(\lambda)\) is \(q\times q\), is called right strongly regular if \[ \|(w_{11} \varepsilon+ w_{12}) (w_{21} \varepsilon+ w_{22})^{-1} \|_\infty< 1 \] for at least one \(p\times q\)-function \(\varepsilon\) which is analytic and contractive in \(\Omega_+\). Characterizations of right strongly regular \(J\)-inner functions in terms of the matricial Muckenhoupt condition were obtained in [\textit{D. A. Arov} and \textit{H. Dym}, J. Funct. Anal. 181, 227--299 (2001; Zbl 0980.47014)]. A positive semidefinite matrix function \(\Delta(\lambda)\) is said to satisfy the matrix Muckenhoupt condition if \[ \sup_I \|(A_I(\Delta))^{1/2} (A_I (\Delta^{-1}))^{1/2}\|< \infty, \] where \(A_I(\Delta)\) is the average of \(\Delta\) of a finite subinterval \(I\) of the boundary of \(\Omega_+\) having positive length. In the paper under review, different characterizations are obtained. One of the main results is as follows: A \(\left[ \begin{smallmatrix} I_p &0\\ 0 &-I_q \end{smallmatrix} \right]\)-inner function \(U\) is right strongly regular if and only if \(\left[ \begin{smallmatrix} I_p &-(w_{22}^{-1} w_{21})^*\\ -w_{22}^{-1} w_{21} &-I_q \end{smallmatrix} \right]\) satisfies the matrix Muckenhoupt condition. This result is further extended to general signature matrices. Analogous results are obtained for the dual notion of left strongly regular \(J\)-inner functions, and for the class of \(\gamma\)-generating matrices; the latter class was introduced and investigated in [\textit{D. Z. Arov}, J. Sov. Math. 52, 3487--3491 (1990; Zbl 0718.41004); ibid. 52, 3421--3425 (1990; Zbl 0900.41002)].
scientific article; zbMATH DE number 1917781