Operator theoretic approach to orthogonal polynomials on an arc of the unit circle

Анотація

We study the probability measures on the unit circle and the multiplication operators acting on appropriate L² spaces. When such a measure does not satisfy the Szegö condition, orthonormal polynomials form an orthonormal basis in this Hilbert space. The multiplication operator can be represented by an upper Hessenberg matrix. The main result concerns certain infinite-dimensional perturbations of the "constant" Hessenberg matrix which have a finite number of eigenvalues off the essential spectrum.