A tangential interpolation problem on the distinguished boundary of the polydisk for the Schur--Agler class
It is now well understood that the interpolation theory of Nevanlinna-Pick type in several complex variables faces serious obstructions. The optimal way to circumvent these difficulties was proposed by \textit{J. Agler} in [Oper. Theory, Adv. Appl. 48, 47-66 (1990; Zbl 0733.32002)]. He replaced the space of bounded analytic functions in a polydisk by a smaller class, nowadays called the Schur-Agler class, on which interpolation problems can be solved in terms of positivity conditions, exactly as in the single variable case. A great deal of energy was put, by many working mathematicians, into understanding this approach to function theory of several variables. The article under review proves the correct tangential interpolation results for the polydisk, with respect to the Schur-Agler class. The main tool is the realization of Schur-Agler functions as characteristic functions of certain operator colligations. The basic theorems and their proofs parallel the classical single variable situation. This very clearly written article is self-contained and accessible to non-experts. I recommend it to all mathematicians interested in modern operator theory.
characteristic functions of operator colligations
scientific article; zbMATH DE number 1866565
scientific article