Inner Products on Modules and Orthogonalization with Invertible Squares

Abstract

In Chapter 1 the main objects of investigation were orthogonal scalar polynomials. In later chapters we will study matrix polynomials that are orthogonal for an inner product with a matrix-valued weight. The set of all polynomials with r × r matrix coefficients is a module over the C*-algebra of r × r matrices. Moreover, in Section 2.2 Krein’s Theorem was restated with the help of a module over a C*-algebra. Thus we are led to consider inner products on modules over C*-algebras. The values of the inner products will be elements of the C*-algebras.