Nevanlinna matrix estimates without regularity conditions

The Nevanlinna matrix of a half-line Jacobi operator coincides, up to multiplication with a constant matrix, with the monodromy matrix of an associated canonical system. This canonical system is discrete in a certain sense, and is determined by two sequences, called “lengths” and “angles”. We derive new lower and upper estimates for the norm of the monodromy matrix in terms of the lengths and angles, without imposing any restrictions on these sequences. Returning to the Jacobi setting, we show that the order of the Nevanlinna matrix is always greater than or equal to the convergence exponent of the off-diagonal sequence of Jacobi parameters, and obtain a generalisation of a classical theorem of Berezanskii.