Very badly approximable matrix functions (Q2490684)

A matrix valued function \(\Phi\) defined on the unit circle \({\mathcal T}\) with values in the space \(M_{m,n}\) of \(m \times n\) matrices is called badly approximable if \[ \| \Phi\| _{L^{\infty}} = \inf{\{\| \Phi - F\| _{L^{\infty}}: F \in H^{\infty}(M_{m,n})\}}, \quad \text{ where } \| \Phi\| _{L^{\infty}} \overset{def}{=} \text{ ess } \sup_{\zeta \in {\mathcal T}}{\| \Phi(\zeta)\| _{M_{m,n}}} . \] For a matrix \(A\) and for \(j \geq 0\), let \(s_{j}(A)\), denote the distance from \(A\) to the set of matrices of rank \(j\). The number \(s_{j}(A)\) is called a singular value of \(A\). For \(\Phi \in L^{\infty}(M_{m,n})\) and for \(0 \leq j \leq \min{\{m,n\}} - 1\), define \[ \Omega_{0} = \{F \in H^{\infty}(M_{m,n}): F \text{ minimizes } t_{0} = \text{ess} \sup_{\zeta \in {\mathcal T}} {\| \Phi(\zeta) - F(\zeta)\| \}} \;, \] and, for \(t > 0\), \[ \Omega_{j} = \{F \in \Omega_{j-1}: F \text{ minimizes } t_{j} = \text{ess} \sup_{\zeta \in {\mathcal T}} {s_{j}(\Phi(\zeta) - F(\zeta))} \} . \] Functions in the set \(\Omega_{\min\{m,n\}-1} = \bigcap_{j>0} \Omega_{j}\) are called superoptimal approximations of \(\Phi\), and \(\Phi\) is called very badly approximable if the zero function is a superoptimal approximation of \(\Phi\). The numbers \(t_{j}\) (used to define the sets \(\Omega_{j}\)) are called superoptimal singular values of \(\Phi\). The function \(\Phi\) is called an admissible function if the essential norm \(\| H_{\Phi}\| _{e}\) of the Hankel operator \(H_{\Phi}\) is less than the smallest superoptimal singular value of \(\Phi\). If \(A\) is an \(m \times n\) matrix and \(s\) is a singular value of \(A\), a non-zero vector \(x \in \mathbb C^{n}\) is called a Schmidt vector corresponding to \(s\) if \(A^{*}Ax = s^{2}x\). For the function \(\Phi\), the set \({\mathcal S}^{(\sigma)}(\zeta)\) is the linear span of all Schmidt vectors of \(\Phi(\zeta)\) that correspond to the singular values greater than or equal to \(\sigma\). The first author and \textit{N. J. Young} [J. Funct. Anal. 120, 300--343 (1994; Zbl 0808.47011)] and \textit{R. B. Alexeev} and the first author [Indiana Univ. Math. J. 40, 1247--1285 (2000; Zbl 0996.47017)] gave full characterizations of very badly approximable functions, using a special factorization, but this factorization is not easy to apply. In the current paper, the authors give some different characterizations of badly and very badly approximable functions using some more straightforward criteria. Two of the theorems are as follows. Theorem 4.1: If \(\Phi\) is an admissible very badly approximable matrix function in \(L^{\infty}(M_{m,n})\), then \(\Phi\) satisfies the condition \[ \begin{multlined} \text{for each } \sigma > 0 \text{ the analytic family of subspaces } {\mathcal S}_{\Phi}^{(\sigma)} \text{ is spanned by } \text{finitely many}\\ \text{ functions in Ker } T_{\Phi}, \text{ where } T_{\Phi} \text{ is the Toeplitz operator} \text{associated with } \Phi ,\end{multlined}\tag{\(*\)} \] and, conversely, if \(\Phi\) is an arbitrary function in \(\L^{\infty}(M_{m,n})\) that satisfies the condition (\(*\)), then \(\Phi\) is a very badly approximable function for which the zero function is the only superoptimal approximation. Theorem 6.1: If \(\Phi \in L^{\infty}(M_{m,n})\) such that \(\| H_{\Phi}\| _{e} < \| \Phi\| _{L^{\infty}}\), then \(\Phi\) is a badly approximable function if and only if both (i) \(\| \Phi(\zeta)\| _{M_{m,n}}\) is constant for almost all \(\zeta \in {\mathcal T}\), and (ii) there exists a function \(f \in \text{Ker } T_{\Phi}\) such that \(f(\zeta)\) is a maximizing vector of \(\Phi(\zeta)\) for almost all \(\zeta \in {\mathcal T}\).