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

Carlsson, M. and Perfekt, K.-M. (2019) Nehari's theorem for convex domain Hankel and Toeplitz operators in several variables. International Mathematics Research Notices. ISSN 1073-7928

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To link to this item 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
Refereed:Yes
Divisions:Faculty of Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:85334
Publisher:Oxford University Press

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