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

Carlsson, M. and Perfekt, K.-M. (2021) Nehari's theorem for convex domain Hankel and Toeplitz operators in several variables. International Mathematics Research Notices, 2021 (5). pp. 3331-3361. ISSN 1073-7928 doi: 10.1093/imrn/rnz193

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.

Item Type	Article
URI	https://centaur.reading.ac.uk/id/eprint/85334
Identification Number/DOI	10.1093/imrn/rnz193
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	Oxford University Press
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