Resolvents of Symmetric Operators and the Degenerated Nevanlinna-Pick Problem

Let H be an n-dimensional space, which is equipped with a positive semidefinite inner product with a one-dimensional isotropic part. Consider a symmetric operator S in H with defect index (1, 1). We give a parametrization of the family of S̆traus relations extending S and acting in a Pontryagin space Pc ⊇ H of dimension n+1 and of negative index 1, and a parametrization of the corresponding set of u-resolvents of S. These results are applied to a classical Nevanlinna-Pick interpolation problem for which the Pick matrix is positive semidefinite but singular: We obtain explicit formulas for the solutions of this interpolation problem belonging to the class N1 (see Section 4). 1