Nonisospectral Flows on Self-adjoint, Unitary and Normal Semi-infinite Block Jacobi Matrices

Abstract

The article gives an overview of recent results in the theory of difference-differential lattices generated by various forms of the Lax equation

(t) = Φ(J(t),t) + [J(t),A(J(t),t)] of the following type. It is required that J(t) : l ₂ → l ₂ be able to be mapped into a self-adjoint, or unitary, or normal operator L(t) of multiplication by an independent variable in separable Hilbert space L ²(ℂ,dρ(·,t)). Here dρ is a probability measure with infinite compact support defined on the Borel σ-algebra $\mathfrak{B}$(ℂ) (spectral measure of L(t)).

The article presents an algorithm that solves such lattices via the Inverse Spectral Problem. For the case of unitary J(t), three applications (in terms of the Verblunsky coefficients) are presented.

The results of the article are closely related to the classical theory of Jacobi matrices (Toda lattice, etc.) and the OPUC theory (Schur flows, etc.).