Spectral Nevanlinna-Pick and Carathéodory-Fejér problems for \(n\leq 3\) (Q2909234)

This paper studies two related problems on interpolation by holomorphic functions in the unit disc \(\mathbb D\) with values in the spectral ball \(\Omega_n=\big\{A\in \mathcal M_n\;|\;r(A)<1\big\}\), with \(n=2,3\).NEWLINENEWLINEProblem 1 (Spectral Nevanlinna-Pick problem). Given \(\alpha_1,\dots,\alpha_k\in\mathbb D\) and \(A_1,\dots,A_k\in\Omega_n\), determine whether there exists \(\psi\in\mathcal O(\mathbb D,\Omega_n)\) such that \(\psi(\alpha_j)=A_j\), \(j=1,\dots,k\). NEWLINENEWLINENEWLINENEWLINE Problem 2 (Spectral Carathéodory-Fejér problem in its simplest case). Given \(A\in\Omega_n\) and \(B\in\mathcal M_n\), determine whether there exists \(\psi\in\mathcal O(\mathbb D,\Omega_n)\) such that \(\psi(0)=A\) and \(\psi'(0)=B\).NEWLINENEWLINEBoth problems are reduced to simpler interpolation problems in the symmetrized polydisk \(\mathbb G_n=\{\sigma(A)\;|\;A\in\Omega_n\}\) (the map \(\sigma\) is defined by the formula \(\det(tI-A)=\sum_{j=0}^n(-1)^j \sigma_j(A) t^{n-j}\)). These new problems are presumably simpler because \(\mathbb G_n\) is a bounded hyperconvex domain of dimension \(n\), while \(\Omega_n\) has dimension \(n^2\). Specifically, the authors give precise conditions so that a solution \(\varphi\in\mathcal O(\mathbb D,\mathbb G_n)\) to the interpolation problem \(\varphi(\alpha_j)=\sigma(A_j)\), \(j=1,\dots,n\), can be lifted to a solution of Problem 1, and a map \(\varphi\in\mathcal O(\mathbb D,\mathbb G_n)\) with \(\varphi(0)=\sigma(A)\) can be lifted to a solution of Problem 2. These conditions are expressed in terms of the derivatives of the components of the map \(\varphi\) on the \(\alpha_j\)'s.NEWLINENEWLINEThe proofs are based on a careful separation by cases, depending on whether the target matrices are cyclic, scalar or none of the two. Along the way some interesting remarks are given and connections to other problems are mentioned.