An index formula in connection with meromorphic approximation

Abstract

Let $\Phi$ be a continuous $n\times n$ matrix-valued function on the unit circle $\T$ such that the $(k-1)$th 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 $\mathbb{D}$ (i.e. the McMillan degree of $Q$ in $\mathbb{D}$ is at most $k$) and $Q$ minimizes the essential suprema of singular values $s_{j}((\Phi-Q)(\zeta))$, $j\geq0$, with respect to the lexicographic ordering. For each $j\geq 0$, the essential supremum of $s_{j}((\Phi-Q)(\zeta))$ is called the $j$th superoptimal singular value of $\Phi$ of degree $k$. 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 \[ \ind T_{\Phi-Q}=\dim\ker T_{\Phi-Q}=2k+\dim\mathcal{E}, \] where $\mathcal{E}=\{\xi\in\ker H_{Q}: \|H_{\Phi}\xi\|_{2}=\|(\Phi-Q)\xi\|_{2}\}$ and $H_{\Phi}$ denotes the Hankel operator with symbol $\Phi$. In fact, this result can be extended from continuous matrix-valued functions to the wider class of $k$-\emph{admissible} matrix-valued functions, i.e. essentially bounded $n\times n$ matrix-valued functions $\Phi$ on $\T$ for which the essential norm of the Hankel operator $H_{\Phi}$ is strictly less than the smallest non-zero superoptimal singular value of $\Phi$ of degree $k$.