Unitary interpolants and factorization indices of matrix functions

Details Unitary interpolants and factorization indices of matrix functions Unitary interpolants and factorization indices of matrix functions

2001-01-26

arxiv.org/abs/math/0101224v1

Mathematics

Functional Analysis

20 pages

Scientific paper

For an $n\times n$ bounded matrix function $\Phi$ we study unitary interpolants $U$, i.e., unitary-valued functions $U$ such that $\hat U(j)=\hat\Phi(j)$, $j<0$. We are looking for unitary interpolants $U$ for which the Toeplitz operator $T_U$ is Fredholm. We give a new approach based on superoptimal singular values and thematic factorizations. We describe Wiener--Hopf factorization indices of $U$ in terms of superoptimal singular values of $\Phi$ and thematic indices of $\Phi-F$, where $F$ is a superoptimal approximation of $\Phi$ by bounded analytic matrix functions. The approach essentially relies on the notion of a monotone thematic factorization introduced in [AP]. In the last section we discuss hereditary properties of unitary interpolants. In particular, for matrix functions $\Phi$ of class $H^\be+C$ we study unitary interpolants $U$ of class $QC$.

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