Interpolation theory in sectorial Stieltjes classes and explicit system solutions

We introduce sectorial classes of matrix-valued Stieltjes functions in which we solve the bitangential interpolation problem of Nudelman and Ball-Gohberg-Rodman. We consider also a new type of solutions of Nevanlinna-Pick interpolation problems, so-called explicit system solutions generated by Brodskii-Livsic colligations, and find conditions on interpolation data of their existence and uniqueness. We point out the connections between sectorial Stieltjes classes and sectorial operators, and find out new properties of the classical Nevanlinna-Pick interpolation matrices (in the scalar case). We present in terms of interpolation data the exact formula for the angle of sectoriality of the main operator in the explicit system solution as well as the criterion for this operator to be extremal.The interpolation model for nonselfadjoint matrices is established.