An extremal problem in interpolation theory (Q2716497)

Let \(z_1,\dots,z_n\) be points in the unit disk \({\mathbf D}\) and \(W_1,\dots,W_n\) be complex \(m\times m\) matrices and let \({\mathcal F}\) denote the set of all \(m\times m\) matrix valued functions \(F(z)\) with entries in \(H^\infty\), such that \(F(z_j)=W_j\) (\(j=1,\ldots,n)\). By the matrix version of the classical Nevanlinna-Pick interpolation theorem, there exists a function \(F\in {\mathcal F}\) with \(\|F\|_{\infty}=\max_{z\in{\mathbf D}}\|F(z)\|\leq \rho\) if and only if the Pick matrix NEWLINE\[NEWLINEP_{\rho}=\left(\frac{\rho^2I_m-W_j W_i^*}{1-z_j\overline{z_i}}\right) NEWLINE\]NEWLINE is positive semidefinite. Thus, NEWLINE\[NEWLINE \rho_0:=\min_{P_\rho \geq 0}\rho=\min_{F\in{\mathcal F}} \|F\|_{\infty}. NEWLINE\]NEWLINE The set of all functions \(F\in{\mathcal F}\) with \(\|F(z)\|_{\infty}=\rho_0\) is not empty (in general, it may contain more than one element) and is parametrized by a linear fractional transformation. In the scalar case (\(m=1\)), the function with minimal norm is unique; moreover, it is rational and of McMillan degree not more than \(n-1\). Replacing the operator norm of \(F(z)\) by the spectral radius \(\|F(z)\|_{\text{sp}}\) we come to the spectral interpolation problem: find all functions \(F\in{\mathcal F}\) with \(\|F\|_{\text{sp}}=\sup_{z\in{\mathbf D}}\|F(z)\|_{\text{sp}}\leq \rho\). Although this problem coinsides for \(m=1\) with well known and well studied classical Nevanlinna-Pick problem, the case \(m>1\) turned out to be extremely difficult. For \(m=2\), necessary conditions for the problem to have a solution were obtained in [\textit{J. Agler} and \textit{N. J. Young}, J. Funct. Anal. 161, No. 2, 452-477 (1999; Zbl 0943.47005)] and the complete solution of the two-point spectral interpolation problem was presented in [\textit{J. Agler} and \textit{N. J. Young}, Integral Equations Oper. Theory 37, No. 4, 375-385 (2000; Zbl 1054.47504)]. The author considers the extremal problem about minimal norm solutions in the context of the spectral interpolation problem: does there exists a function \(F_0\in{\mathcal F}\) such that NEWLINE\[NEWLINE \|F_0\|_{\text{sp}}=\inf_{F\in{\mathcal F}} \|F\|_{\text{sp}}?NEWLINE\]NEWLINE The author gives affirmative answers for the cases when \(m=2,3\) and points out the difficulty of extending his result to \(m>3\).