Abstract
We give a new algorithm for the construction of the unique superoptimal analytic approximant of a given continuous matrix-valued function on the unit circle, making use of exterior powers of operators in preference to spectral or {\em Wiener-Masani} factorizations.
Highlights
For Hilbert spaces H, K, we define by L(H, K) the Banach space of bounded linear operators from H to K with the operator norm
We denote by K(H, K) the Banach space of compact linear operators from H to K with the operator norm
Important in the development of the theory was a series of deep papers by Adamyan, Arov and Krein [1, 2] which greatly extend Nehari’s theorem and which apply to matrix-valued functions
Summary
The Nehari problem of approximating an essentially bounded Lebesgue measurable function on the unit circle T by a bounded analytic function on the unit disk D, has been attracting the interest of both pure mathematicians and engineers since the middle of the 20th century. Peller and Young proved some requisite preparatory results on ‘thematic factorizations’, on the analyticity of the minors of unitary completions of inner matrix columns and on the compactness of some Hankel-type operators with matrix symbols These results provided the foundation for their main theorem, namely that if G belongs to H∞(D, Cm×n) + C(T, Cm×n), there exists a unique Q ∈ H∞(D, Cm×n) such that the sequence s∞(G − Q) is lexicographically minimized as Q varies over H∞(D, Cm×n); for this Q, the singular values sj(G(z) − Q(z) are constant almost everywhere for z ∈ T, for j = 0, 1, 2,. Peller and Young’s paper [25] provided the motivation for the study of this problem, where they were able to algebraically characterize the very badly approximable matrix functions of class H∞(D, Cm×n) + C(T, Cm×n) Their results were extended in [23] to the case of matrix functions G for which HG e is less than the smallest non-zero superoptimal singular value of G.
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