Unitary interpolants, factorization indices and infinite Hankel Block matrices (Q790400)

The existence, uniqueness, and construction of unitary \(n\times n\) matrix valued functions \(f(\zeta)=\sum^{\infty}_{j=-\infty}f_ j\zeta^ j\) in Wiener-like algebras on the circle with prescribed matrix Fourier coefficients \(f_ j=\gamma_ j\) for \(j\geq 0\) are studied. In particular, if \(\Sigma | \gamma_ j|<\infty,\) then such an f exists with \(\Sigma | f_ j|<\infty\) if and only if \(\| \Gamma_ 0\| \leq 1,\) where \(\Gamma_ v\) denotes the infinite block Hankel matrix \((\gamma_{j+k+v})\), \(j,k=0,1,..\). acting in the sequence space \(\ell^ 2_ n\). One of the main results is that the nonnegative factorization indices of every such f are uniquely determined by the given data in terms of the dimensions of the kernels of \(1-\Gamma_ v^*\Gamma_ v,\) whereas the negative factorization indices are arbitrary. It is also shown that there is a unique such f if and only if the data forces all the factorization indices to be nonnegative and simple conditions for that and a formula for f in terms of certain Schmidt pairs of \(\Gamma_ 0\) are given. The results depend upon a fine analysis of the structure of the kernels of \(I-\Gamma_ v^*\Gamma_ v,\) and of the one step extension problem of \textit{V. M. Ada\`mjan}, \textit{D. Z. Arov} and \textit{M. G. Krein} [Funkts. Anal. Prilozh. 2, 1-17 (1968; Zbl 0179.467)]. Isometric interpolants for the nonsquare case are also considered.