Abstract/Summary

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 Ξ = (0, 1)d be a d-dimensional cube, and for a distribution f on 2Ξ, consider the Hankel operator Γf (g)(x) = ʃΞ f(x + y)g(y)dy, x ∈ Ξ Then Γf extends to a bounded operator on L2(Ξ) if and only if there is a bounded function b on Rd whose Fourier transform coincides with f on 2Ξ. 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.