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Singular Values of Trilinear Forms

Bernhardsson, Bo LU and Peetre, Jaak LU (2001) In Experimental Mathematics 10(4). p.509-518
Abstract
Let T : H 1 × H 2 × H 3 → C be a trilinear form, where H 1, H 2, H 3 are separable Hilbert spaces. In the hypothesis that at least two of the three spaces are finite dimensional we show that the norm square λ = ∥T∥2 is a root of a certain algebraic equation, usually of very high degree, which we baptize the millennia] equation, because it is an analogue of the secular equation in the bilinear case. More generally, as indicated in the title, we can consider singular values of a trilinear form and their squares too satisfy the same equation. We work out the binary case (all three spacesare two dimensional). Even in this case the situation is complex, so, in the absence of any genuine results, we content ourselves with advancing a number of... (More)
Let T : H 1 × H 2 × H 3 → C be a trilinear form, where H 1, H 2, H 3 are separable Hilbert spaces. In the hypothesis that at least two of the three spaces are finite dimensional we show that the norm square λ = ∥T∥2 is a root of a certain algebraic equation, usually of very high degree, which we baptize the millennia] equation, because it is an analogue of the secular equation in the bilinear case. More generally, as indicated in the title, we can consider singular values of a trilinear form and their squares too satisfy the same equation. We work out the binary case (all three spacesare two dimensional). Even in this case the situation is complex, so, in the absence of any genuine results, we content ourselves with advancing a number of conjectures suggested by computer experiments. Finally, we connect the singular values of a trilinear form with the critical values of an associated family of a one parameter family of bilinear forms. Also here we have to offer mainly only experimental evidence.


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author
organization
publishing date
type
Contribution to journal
publication status
published
subject
in
Experimental Mathematics
volume
10
issue
4
pages
509 - 518
publisher
A K Peters LTD
external identifiers
  • scopus:0035566386
ISSN
1944-950X
DOI
10.1080/10586458.2001.10504670
language
English
LU publication?
yes
id
03f84a25-013c-4bca-b72a-45e0f7cdd444
date added to LUP
2017-11-27 09:13:36
date last changed
2018-05-29 10:45:41
@article{03f84a25-013c-4bca-b72a-45e0f7cdd444,
  abstract     = {Let T : H 1 × H 2 × H 3 → C be a trilinear form, where H 1, H 2, H 3 are separable Hilbert spaces. In the hypothesis that at least two of the three spaces are finite dimensional we show that the norm square λ = ∥T∥2 is a root of a certain algebraic equation, usually of very high degree, which we baptize the millennia] equation, because it is an analogue of the secular equation in the bilinear case. More generally, as indicated in the title, we can consider singular values of a trilinear form and their squares too satisfy the same equation. We work out the binary case (all three spacesare two dimensional). Even in this case the situation is complex, so, in the absence of any genuine results, we content ourselves with advancing a number of conjectures suggested by computer experiments. Finally, we connect the singular values of a trilinear form with the critical values of an associated family of a one parameter family of bilinear forms. Also here we have to offer mainly only experimental evidence.<br/><br/> <br/>},
  author       = {Bernhardsson, Bo and Peetre, Jaak},
  issn         = {1944-950X},
  language     = {eng},
  number       = {4},
  pages        = {509--518},
  publisher    = {A K Peters LTD},
  series       = {Experimental Mathematics},
  title        = {Singular Values of Trilinear Forms},
  url          = {http://dx.doi.org/10.1080/10586458.2001.10504670},
  volume       = {10},
  year         = {2001},
}