Some solutions to the matrix equation for three point Nevanlinna-Pick interpolation on the bidisc

AbstractSome explicit solutions to the 3 × 3 case of Agler's matrix equation for Nevanlinna-Pick interpolation on the bidisc are provided. Agler showed there exists a holomorphic function bounded by 1 on the bidisc D2 which maps n prescribed points in D2 to n prescribed points in D if and only if there exists a pair of n × n positive semidefinite matrices satisfying a certain matrix equation. We show there exists a solution to Agler's equation in which one matrix has a row and column of zeros if and only if an explicit set of inequalities that depend on the data are satisfied. A solution of this form is also equivalent to the existence of an interpolating function which is constant with respect to one coordinate of one bidisc data point. The best possible bounds on diagonal elements of solutions to Agler's 2 × 2 matrix equation are also provided