Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials

Aptekarev, Alexander; Denisov, Sergey; Yattselev, Maxim

doi:10.1090/tran/7959

Self-adjoint Jacobi matrices on trees and multiple orthogonal polynomials
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by Alexander I. Aptekarev, Sergey A. Denisov and Maxim L. Yattselev PDF

Trans. Amer. Math. Soc. 373 (2020), 875-917 Request permission

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 $\mathbb {Z}_+$ to a higher dimension. We illustrate the importance of this connection by proving ratio asymptotics for MOPs using methods of operator theory.

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