NDLI: An Index Formula in Connection with Meromorphic Approximation

Content Provider	Springer Nature Link
Author	Condori, Alberto A.
Copyright Year	2012
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 $$\Phi $$ .
Starting Page	1787
Ending Page	1805
Page Count	19
File Format	PDF
ISSN	16618254
Journal	Complex Analysis and Operator Theory
Volume Number	7
Issue Number	6
e-ISSN	16618262
Language	English
Publisher	Springer Basel
Publisher Date	2012-07-18
Publisher Place	Basel
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	Nehari–Takagi problem Hankel and Toeplitz operators Best approximation Badly approximable matrix-valued functions Superoptimal approximation Operator methods in interpolation, moment and extension problems Toeplitz operators, Hankel operators, Wiener-Hopf operators Spaces of vector- and operator-valued functions Mathematics Operator Theory Analysis
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Theory and Mathematics Computational Mathematics

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