Unbounded banded matrices with positive bidiagonal factorization and mixed-type multiple orthogonal polynomials

Abstract: A spectral Favard theorem is proved for semi-infinite banded matrices admitting a positive bidiagonal factorization, without assuming boundedness of the associated operator, thus covering both the bounded and unbounded settings. The result yields a matrix-valued spectral measure and an explicit spectral representation of the matrix powers in terms of the associated mixed-type multiple orthogonal polynomials. The argument follows the constructive truncation scheme: principal truncations are oscillatory, hence have simple positive spectra, and a suitable choice of initial conditions ensures positivity of the Christoffel coefficients and of the resulting discrete matrix-valued measures supported at the truncation eigenvalues. The main difficulty is the passage to the limit of these discrete measures beyond the bounded case. This is resolved by combining the available Gaussian quadrature structure with a Helly-type compactness argument, leading to a limiting matrix-valued measure and completing the spectral theorem. The role of normality (maximal degree pattern) for the mixed-type families is also addressed.