Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials

Abstract

We consider a set of measures on the real line and the corresponding system of multiple orthogonal polynomials (MOPs) of the first and second type. Under some very mild assumptions, which are satisfied by Angelesco systems, we define self-adjoint Jacobi matrices on certain rooted trees. We express their Green's functions and the matrix elements in terms of MOPs. This provides a generalization of the well-known connection between the theory of polynomials orthogonal on the real line and Jacobi matrices on Z + to higher dimension. We illustrate importance of this connection by proving ratio asymptotics for MOPs using methods of operator theory. Contents arXiv:1806.10531v1 [math.CA] 27 Jun 2018where Y (p) has d children each corresponding to the index i ∈ {1, . . . , d}.First, we claim that J κ, N → J κ, c in the strong operator sense, i.e.,for every fixed f ∈ 2 (V). Indeed, let χ |X|<ρ be the characteristic function of the ball in T with center at O and radius ρ. Given any > 0, there is ρ such thatSince coefficients {a n,j } and {b n,j } are uniformly bounded, we haveuniformly in N . Having and ρ fixed, we getby our assumptions (4.19). This proves our claim. Next, the Second Resolvent Identity from perturbation theory of operators givesfor z ∈ C\R. Since (J ej , c − z) −1 | Im z| −1 by the Spectral Theorem, we can take | N | → ∞ and use the above claim to obtain (4.21) lim

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

“…The following theorem provides the connection between operators L plq c and J κ, N . It is stated in [9] for c P p0, 1q and is a simple consequence of the results of [42]. Its extension to c P t0, 1u was obtained in [10].…”

“…In this part of the paper, we consider Angelesco system, as in Part 3, see (3.0.1), in the case when supp µ i " ∆ i , dµ i pxq " µ 1 i pxqdx, µ i pxq ą 0, x P ∆ i , and µ 1 i pxq is a restriction of an analytic function defined around ∆ i . This situation was studied in great detail in [9] and [10], see also [42]. In particular, it was proved that J κ, N converges to a limiting operator L piq c when N goes to infinity along the ray N c " n : n i " c i | n| `op| n|q, i P t1, 2u ( , pc 1 , c 2 q " pc, 1 ´cq, c P r0, 1s.…”

“…In the current paper, we study the spectrum and spectral decomposition in a more general situation. We focus on the case of two measures only and address several questions that were left open in [9].…”

A Glazman-Povzner-Wienholtz Theorem on graphs

“…Remark 6.3 (Jacobi matrices on graphs). Sometimes the second order difference expression (6.3) is called a Jacobi matrix on a graph (cf., e.g., [3], [4]). Indeed, if V = Z ≥0 and b is a path graph over Z ≥0 , that is, b(n, m) > 0 exactly when |n − m| = 1, then (6.3) can be written as a Jacobi (tri-diagonal) matrix…”

Christoffel functions for multiple orthogonal polynomials

“…First of all, in view of [1,Remark A.11] the hypotheses of Proposition 3.2 are satisfied. Hence, the corresponding matrix satisfies (NDB).…”