Rank one perturbations of Jacobi matrices with mixed spectra

Let A be a self‐adjoint operator and φ its cyclic vector. In this work we study spectral properties of rank one perturbations of AA θ = A + θ 〈 φ , ·〉 φin relation to their dependence on the real parameter θ . We find bounds on averages of spectral measures for semi‐infinite Jacobi matrices and give criteria which guarantee existence of mixed spectral types for θ in a set of positive Lebesgue measure. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)