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 $\Phi$.