Orthogonal systems and convolution operators
The monograph by R. Ellis and I. Gohberg is devoted to orthogonal polynomials on the unit circle with indefinite inner product and their generalizations. The authors start with the Krein theorem concerning the distribution of zeros of polynomials orthogonal with respect to nonpositive weight function and their continuous analogues. They explore various generalizations and extensions of the theory and consider a more general problem of orthogonalization in modules over \(C^*\)-algebras. One of the main features of the book is the interplay between orthogonal polynomials and generalizations on the one hand, and operator theory on the other hand, especially the theory of Toeplitz matrices, Wiener-Hopf operators, and Fredholm operators and their indices. Various inertia theorems and extended Rouché's theorems come in frequently. In the same way as orthogonal polynomials are connected with Toeplitz matrices, the continuous analogues of orthogonal polynomials are connected with convolution operators on a finite interval. The study of such operators with a focus on invertibility conditions and an inversion formula is one of the key subjects of the book. The book is of interest to both engineers and specialists in analysis.
scientific article; zbMATH DE number 1860219
scientific article