Singular Values of Trilinear Forms
(2001) In Experimental Mathematics 10(4). p.509518 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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Please use this url to cite or link to this publication:
https://lup.lub.lu.se/record/03f84a25013c4bcab72a45e0f7cdd444
 author
 Bernhardsson, Bo ^{LU} and Peetre, Jaak ^{LU}
 organization
 publishing date
 2001
 type
 Contribution to journal
 publication status
 published
 subject
 in
 Experimental Mathematics
 volume
 10
 issue
 4
 pages
 509  518
 publisher
 A K Peters
 external identifiers

 scopus:0035566386
 ISSN
 1944950X
 DOI
 10.1080/10586458.2001.10504670
 language
 English
 LU publication?
 yes
 id
 03f84a25013c4bcab72a45e0f7cdd444
 date added to LUP
 20171127 09:13:36
 date last changed
 20200113 00:15:50
@article{03f84a25013c4bcab72a45e0f7cdd444, 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 = {1944950X}, language = {eng}, number = {4}, pages = {509518}, publisher = {A K Peters}, series = {Experimental Mathematics}, title = {Singular Values of Trilinear Forms}, url = {http://dx.doi.org/10.1080/10586458.2001.10504670}, doi = {10.1080/10586458.2001.10504670}, volume = {10}, year = {2001}, }