Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: An index formula in connection with meromorphic approximation   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 $k$.

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The categories of this paper are: bayes-an
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The categories of this paper are: atom-ph
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The categories of this paper are: cond-mat
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The categories of this paper are: dg-ga
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The categories of this paper are: hep-ex
1
The categories of this paper are: cs
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The categories of this paper are: gr-qc
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The categories of this paper are: nucl-ex
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The categories of this paper are: plasm-ph
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The categories of this paper are: chao-dyn
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The categories of this paper are: acc-phys
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The categories of this paper are: nucl-th
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The categories of this paper are: math-ph
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The categories of this paper are: q-fin
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The categories of this paper are: chem-ph
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The categories of this paper are: adap-org