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 $${\Gamma }_f(g)(x)={\int }_{\Xi }f(x+y)g(y)dy,x\in \Xi .$$ 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.

arXiv comment20 pages. Minor corrections in Section 7. Brief historical overview added