Orthogonal Systems and Convolution Operators

The main concern of this book is the distribution of zeros of polynomials that are othogonal on the unit circle with respect to an indefinite weighted scalar or inner product. The first theorem of this type, proved by M.G. Krein, was a far-reaching generalisation of G. Szego’s result for the positive definite case. A continuous analogue of that theorem was proved by Krein and H. Langer. These results, as well as many generalisations and extensions are thoroughly treated in this work. A unifying theme is the general problem of orthogonalization with invertible squares in modules over C*-algebras. Particular modules that are considered in detail include modules of matrices, matrix polynomials, matrix-valued functions, linear operators and others. One of the main features of this book is the interplay between orthogonal polynomials and their generalizations on the one hand, and operator theory, especially the theory of Toeplitz matrices and operators, and Fredholm and Wiener-Hopf operators, on the other hand. The work should be of interest to both engineers and specialists in analysis.