An Index Formula in Connection with Meromorphic Approximation

Abstract

Let \(\Phi \) be a continuous \(n\times n\) matrix-valued function on the unit circle \(\mathbb T \) such that the \((k-1)\)st singular value of the Hankel operator with symbol \(\Phi \) is greater than the \(k\)th singular value. In this case, it is well-known that \(\Phi \) has a unique superoptimal meromorphic approximant \(Q\) in \(H^{\infty }_{(k)}\); that is, \(Q\) has at most \(k\) poles in the unit disc \(\mathbb D \) (in the sense that the McMillan degree of \(Q\) in \(\mathbb D \) is at most \(k\)) and \(Q\) minimizes the essential suprema of singular values \(s_{j}\left((\Phi -Q)(\zeta )\right)\!, j\ge 0\), with respect to the lexicographic ordering. For each \(j\ge 0\), the essential supremum of \(s_{j}\left((\Phi -Q)(\zeta )\right)\) is called the \(j\)th superoptimal singular value of degree \(k\) of \(\Phi \). We prove that if \(\Phi \) has \(n\) non-zero superoptimal singular values of degree \(k\), then the Toeplitz operator \(T_{\Phi -Q}\) with symbol \(\Phi -Q\) is Fredholm and has index $$ \mathrm{ind}T_{\Phi -Q}=\dim \ker T_{\Phi -Q}=2k+\dim \mathcal E , $$ where \(\mathcal E =\{ \xi \in \ker H_{Q}: \Vert H_{\Phi }\xi \Vert _{2}=\Vert (\Phi -Q)\xi \Vert _{2}\}\) and \(H_{\Phi }\) denotes the Hankel operator with symbol \(\Phi \). This result can in fact be extended from continuous matrix-valued functions to the wider class of \(k\)-admissible matrix-valued functions, i.e. essentially bounded \(n\times n\) matrix-valued functions \(\Phi \) on \(\mathbb T \) for which the essential norm of the Hankel operator \(H_{\Phi }\) is strictly less than the smallest non-zero superoptimal singular value of degree \(k\) of \(\Phi \).