Skip to main content

Lund University Publications

LUND UNIVERSITY LIBRARIES

Nehari's Theorem for Convex Domain Hankel and Toeplitz Operators in Several Variables

Carlsson, Marcus LU and Perfekt, Karl Mikael LU (2021) In International Mathematics Research Notices 2021(5). p.3331-3361
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 = (0, 1)d be a d-dimensional cube, and for a distribution f on 2, consider the Hankel operator f (g)(x) = λ f (x + y)g(y) dy, x Then λf extends to a bounded operator on L2 if and only if there is a bounded function b on Rd whose Fourier transform coincides with f on 2. This special case has an immediate application in matrix extension theory: every finite multilevel block Toeplitz matrix can be boundedly extended to an infinite multilevel block Toeplitz matrix. In... (More)

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 = (0, 1)d be a d-dimensional cube, and for a distribution f on 2, consider the Hankel operator f (g)(x) = λ f (x + y)g(y) dy, x Then λf extends to a bounded operator on L2 if and only if there is a bounded function b on Rd whose Fourier transform coincides with f on 2. This special case has an immediate application in matrix extension theory: every finite multilevel block Toeplitz matrix can be boundedly extended to an infinite multilevel block Toeplitz matrix. In particular, block Toeplitz operators with blocks that are themselves Toeplitz can be extended to bounded infinite block Toeplitz operators with Toeplitz blocks.

(Less)
Please use this url to cite or link to this publication:
author
and
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
International Mathematics Research Notices
volume
2021
issue
5
pages
31 pages
publisher
Oxford University Press
external identifiers
  • scopus:85126309075
ISSN
1073-7928
DOI
10.1093/imrn/rnz193
language
English
LU publication?
yes
id
9fce41c0-dc52-4945-8306-065062cc543e
date added to LUP
2022-05-02 15:21:23
date last changed
2023-05-10 14:07:05
@article{9fce41c0-dc52-4945-8306-065062cc543e,
  abstract     = {{<p>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 = (0, 1)d be a d-dimensional cube, and for a distribution f on 2, consider the Hankel operator f (g)(x) = λ f (x + y)g(y) dy, x Then λf extends to a bounded operator on L2 if and only if there is a bounded function b on Rd whose Fourier transform coincides with f on 2. This special case has an immediate application in matrix extension theory: every finite multilevel block Toeplitz matrix can be boundedly extended to an infinite multilevel block Toeplitz matrix. In particular, block Toeplitz operators with blocks that are themselves Toeplitz can be extended to bounded infinite block Toeplitz operators with Toeplitz blocks.</p>}},
  author       = {{Carlsson, Marcus and Perfekt, Karl Mikael}},
  issn         = {{1073-7928}},
  language     = {{eng}},
  number       = {{5}},
  pages        = {{3331--3361}},
  publisher    = {{Oxford University Press}},
  series       = {{International Mathematics Research Notices}},
  title        = {{Nehari's Theorem for Convex Domain Hankel and Toeplitz Operators in Several Variables}},
  url          = {{http://dx.doi.org/10.1093/imrn/rnz193}},
  doi          = {{10.1093/imrn/rnz193}},
  volume       = {{2021}},
  year         = {{2021}},
}