On the spectra of infinite Hessenberg and Jacobi matrices

The probability measures on the unit circle are studied in conjunction with multiplication operators acting in appropriate Hilbert spaces. The measure with constant reflection coefficients and the corresponding operator are treated as unperturbed objects. Under certain perturbations of this measure it is shown that the support of perturbed measure contains the support of the original one. More generally, we evaluate the size of gaps inside the support of the perturbed measure. The similar results pertaining to Jacobi and banded matrices are also under consideration.