Inversion of quasitriangular block Toeplitz matrices (Q1187513)

Let \(A^ +(z)\) be an \(s\times s\) matrix function analytic at \(z=0\) and with determinant \(\not\equiv 0\). For integers \(n,q>0\), let \(T_ n(q)\) be the block Toeplitz matrix \((A_{i-j})_{ikj=0}^{n-1}\), where \(A_ i\) is the \(s\times s\) matrix coefficient of \(z^ i\) in the Laurent expansion for \(z^{-q}A^ +(z)\). The author derives explicit formulae for the determinant and inverse of \(T_ n(q)\). Since the case where \(A^ +(0)\) is nonsingular is known, it is assumed that \(A^ +(0)\) is singular but not zero. An essential step in the proofs is to show that \(T_ n(q)\) is non- singular if, and only if, a certain linear operator \(W_{q,m}(n)\) of dimension \((q+m)s\) is invertible; here, \(m\) is the smallest integer such that \(z^ m A^ +(z)^{-1}\) is analytic at \(z=0\). The formula for \(\text{det }T_ n(q)\) is a simple expression in the determinants of \(W_{q,m}(n)\), \(W_{q,m}(0)\), \(W_{m,m}(0)\) and a further linear operator \(C\) of dimension \((m+1)s\).