**August 18, 2020.** Two preprints were submitted lately. One is “*Jacobi matrices on trees generated by Angelesco system: asymptotics of coefficients and essential spectrum” *with A. Aptekarev and M. Yatselev. For Angelesco system with two real-analytic weights, we performed the Riemann-Hilbert analysis to obtain asymptotics of polynomials of the first and second type when the multi-index goes to infinity in any direction, including the marginal ones. That was used to characterize the right limits and the essential spectrum of the corresponding Jacobi matrix on the binary tree.

The second preprint *“Spectral theory of Jacobi matrices on trees whose coefficients are generated by multiple orthogonality“* with M. Yattselev finishes the program initiated in the original paper

*A. Aptekarev, S. Denisov, and M. Yattselev, Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials, Trans. Amer. Math. Soc., Vol. 373, N2, 2020, 875-917*

now published in Transactions of AMS. When two measures of orthogonality are involved, we study Jacobi matrixes generated by MOPs of the first and of the second types. Those of the second type give rise to a Jacobi matrix on the finite binary tree, which is self-adjoint in indefinite metric (it is sign-definite for Angelesco system and is non-sign-definite for Nikishin system). We find the spectrum and construct the basis of eigenvectors each of which can be written in terms of these MOPs. The analysis for MOPs of the first type is more involved since the associated Jacobi matrix is defined on the infinite binary tree. For Angesleco system, this Jacobi matrix is self-adjoint in the standard metric and we find the convenient decomposition of the Hilbert space into the orthogonal sum of cyclic subspaces. Each subspace is described by a generalized eigenvector which is written via the MOPs of the first type. The spectral measures of generators are found explicitly thus making the complete analysis of the spectral type possible. In short, this paper puts the connection between MOP and the operator theory on the solid ground. Unlike for one-dimensional theory, the feature of multi-dimensional case is that often times the theory of self-adjoint operators in indefinite metric (in Krein spaces) starts to play the role.

***********************************************************************

**June 18, 2018.** In the recent preprint with A. Aptekarev and M. Yattselev, we found the missing link between the spectral theory of self-adjoint operators and the theory of multiple orthogonal polynomials. Recall that given a measure on the real line with compact support, we can construct a sequence of orthonormal polynomials which satisfy three-term recurrence that defines one-sided Jacobi matrix. This matrix is actually a self-adjoint operator in for which many standard quantities of operator theory (e.g., Green’s function, spectral measures, etc) can be computed through the measure we started with and the associated orthogonal polynomials. Conversely, given a bounded self-adjoint Jacobi matrix, we can uniquely find the measure of orthogonality that generates it. This correspondence proved to be very useful both in spectral theory and approximation theory (however, to my knowledge never played a crucial role in proving deeper analytical results). The multiple orthogonal polynomials (MOPs) can be of two types and they are defined given measures through some orthogonality conditions (see the paper for details). These polynomials depend on multi-index of dimension and satisfy recurrence relations on the integer lattice. It turns out that, to define the corresponding operator, one has to untwine these recurrences to the rooted homogeneous tree. The resulting operator is a self-adjoint Jacobi matrix which is defined on that tree. We obtained, among other things, the formulas connecting the Green’s function to MOPs and found its spatial asymptotics for analytic weights using matrix Riemann-Hilbert analysis. The usefulness of this connection is illustrated by, e.g., reproving some known results.