Spectral theory of Jacobi matrices on trees whose coefficients are generated by multiple orthogonality

Jacobi matrices on trees whose coefficients are generated by multiple orthogonal polynomials will be introduced. The definition is such that the matrices are self-adjoint when multiple orthogonal polynomials come from Angelesco systems of measures (but is not restricted to this case). Full description of eigenvalues and eigenvectors is obtained for finite Jacobi matrices under very mild assumption that "consecutive" multiple orthogonal polynomials don’t have common zeroes. For infinite Jacobi matrices of Angelesco systems Hilbert space decomposition into an orthogonal sum of cyclic subspaces is obtained. For each of these subspaces, generators and the generalized eigenfunctions are found and written in terms of the orthogonal polynomials. This leads to the identification of the spectrum and the spectral type of such matrices. It also shown that Jacobi matrices of Nikishin systems are neither self-adjoint nor bounded. Joint work with Sergey Denisov.