On spectral tangential Nevanlinna-Pick interpolation (Q1173802)

Given \(z_ 1,z_ 2,\dots,z_ n\) in the unit disc of \(C\) and \(F_ 1,F_ 2,\dots,F_ n\in M_ n(C)\), the classical Nevanlinna-Pick problem consists in finding necessary and sufficient conditions for the existence of an analytic function \(F: D\to M_ N(C)\) such that \(F(z_ j)=F_ j\) \((j=1,2,\dots,n)\) and \(\| F\|_ \infty\leq 1\). \textit{I. P. Fedcina} [Akad. Nauk Arjan. SSR, Dokl. 60, 37-42 (1975; Zbl 0324.41001)] studied the so-called tangential Nevanlinna-Pick problem: given \(u_ j\), \(v_ j\) (non-zero vectors of \(C^ N\)) to find necessary and sufficient conditions for the existence of an analytic function \(F: D\to M_ N(C)\) such that \(F(z_ j)u_ j=v_ j\) \((j=1,2,\dots,n)\) and \(\| F\|_ \infty\leq 1\). In another paper ``A spectral commutant lifting theorem'', Trans. Am. Math. Soc. (to appear) the authors gave necessary and sufficient conditions for the existence of an interpolating \(F\), for the classical problem, whose spectral radius is \(\leq 1\). In the paper under review the authors prove a similar result for the tangential problem. In both papers the results are deduced as consequences of two general spectral commutant lifting theorems. An interesting feature is an explicit algorithm for finding the optimal interpolants.