Nehari's theorem for convex domain Hankel and Toeplitz operators in several variables

TL;DR

This paper extends Nehari's theorem to integral Hankel and Toeplitz operators on convex polytopes in multiple variables, providing a boundedness criterion and applications to matrix extension theory.

Contribution

It generalizes Nehari's theorem to several variables and convex domains, establishing a boundedness criterion for Hankel and Toeplitz operators in this setting.

Findings

Nehari's theorem is proved for convex polytopes in multiple variables.

A boundedness criterion for Hankel operators on the Paley-Wiener space is established.

Finite multi-level block Toeplitz matrices can be extended to bounded infinite block Toeplitz matrices.

Abstract

We prove Nehari's theorem for integral Hankel and Toeplitz operators on simple convex polytopes in several variables. A special case of the theorem, generalizing the boundedness criterion of the Hankel and Toeplitz operators on the Paley-Wiener space, reads as follows. Let $\Xi = (0,1)^d$ be a $d$-dimensional cube, and for a distribution $f$ on $2\Xi$, consider the Hankel operator $$\Gamma_f (g)(x)=\int_{\Xi} f(x+y) g(y) \, dy, \quad x \in\Xi.$$ Then $\Gamma_f$ extends to a bounded operator on $L^2(\Xi)$ if and only if there is a bounded function $b$ on $\mathbb{R}^d$ whose Fourier transform coincides with $f$ on $2\Xi$. This special case has an immediate application in matrix extension theory: every finite multi-level block Toeplitz matrix can be boundedly extended to an infinite multi-level block Toeplitz matrix. In particular, block Toeplitz operators with blocks which are themselves Toeplitz, can be extended to bounded infinite block Toeplitz operators with Toeplitz blocks.