Nehari's Theorem for Convex Domain Hankel and Toeplitz Operators in Several Variables
(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)
- author
- Carlsson, Marcus LU and Perfekt, Karl Mikael LU
- organization
- publishing date
- 2021-03
- 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}}, }