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

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.