Fixed Points of Pick and Stieltjes functions: A Linear Algebraic Approach

The functions analytic in the upper half-plane and mapping the upper-half plane into itself (the so-called Pick functions) play a prominent role in several branches of mathematics. In this thesis we study fixed points of such functions. It is known that a Pick-class function different from the identity map can have at most one fixed point in the upper-half plane. However, it may have many (even infinitely many) appropriately defined boundary fixed points. We establish relations between the values of the derivative of a Pick function at these fixed points. Similar questions are considered in the context of Stieltjes-class functions which, in addition, are analytic on the positive half-axis and map this half-axis into itself