On factorization of trigonometric polynomials
In the present paper, using only ideas from elementary operator theory, a new proof of the operator version of the Fejer-Riesz theorem is given, and some of the ramifications of the ideas of the proof are studied. Starting with a sketch of some basic results on Schur complements and factorization, some simpler proofs are given. Moreover, Schur complements are applied to the study of doubly infinite Toeplitz operators with operator coefficients. A proof of the operator Fejer-Riesz theorem is given in the second section of the paper, and in the third one an explicit construction of the outer factorization is given. Also, several interesting problems are raised by this construction. In the fourth section, a characterization of outer factorizations and extremals is presented, and in the fifth one it is shown how the proof given of the operator Fejer-Riesz theorem can be extended to study the factorization of trigonometric polynomials in several variables. Since even in this case, there are numerous examples of positive polynomials which cannot be factored in terms of polynomials, in the sixth section some sufficient conditions for factorization of multivariate trigonometric polynomials are given. In the last section of the paper, a number of families of operators are constructed which lie somewhere between the contractions and numerical radius contractions, and which are also described via positive trigonometric polynomials.
scientific article; zbMATH DE number 2111520