Jost asymptotics for matrix orthogonal polynomials on the real line

We obtain matrix-valued Jost asymptotics for block Jacobi matrices under an L1-type condition on Jacobi parameters, and give a necessary and sufficient condition for an analytic matrix-valued function to be the Jost function of a block Jacobi matrix with exponentially converging parameters. This establishes the matrix-valued analogue of Damanik-Simon-II paper [6]. The above results allow us to fully characterize the matrix-valued Weyl-Titchmarsh m-functions of block Jacobi matrices with exponentially converging parameters

L^1-spectrum of Banach space valued Ornstein-Uhlenbeck operators

We characterize the L^1(E,μ_∞)-spectrum of the Ornstein–Uhlenbeck operator Lf(x) = (1/2)TrQD^(2) + , where μ_∞ is the invariant measure for the Ornstein–Uhlenbeck semigroup generated by L. The main result covers the general case of an infinite-dimensional Banach space E under the assumption that the point spectrum of A* is nonempty and extends several recent related results

Finite range perturbations of finite gap Jacobi and CMV operators

Necessary and sufficient conditions are presented for a measure to be the spectral measure of a finite range perturbation of a Jacobi or CMV operator from a finite gap isospectral torus. The special case of eventually periodic operators solves an open problem of Simon [25, D.2.7]. We also solve the inverse resonance problem: it is shown that an operator is completely determined by the set of its eigenvalues and resonances, and we provide necessary and sufficient conditions on their configuration for such an operator to exist.Comment: this is the full-details version of the paper accepted for publication in Advances in Mathematics 30 page