×

Singular values of trilinear forms. (English) Zbl 1008.47002

The authors address the following problem. Let \(T:{\mathcal H}_1\times{\mathcal H}_2\times{\mathcal H}_3\to{\mathbb C}\) be a trilinear form, where \({\mathcal H}_1,{\mathcal H}_2,{\mathcal H}_3\) are separable Hilbert spaces. Find the norm \(\|T\|=\sup_{\|x\|\leq 1, \|y\|\leq 1,\|z\|\leq 1}|T(x,y,z)|\). If at least two of the Hilbert spaces are finite dimensional, the authors show that \(\|T\|^2\) is a root of a certain algebraic equation (called the millennial equation). More generally, the authors consider singular values of trilinear forms and work out the case when the spaces are all two dimensional. Some conjectures are advanced, suggested by computer experiments.

MSC:

47A07 Forms (bilinear, sesquilinear, multilinear)
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
47A75 Eigenvalue problems for linear operators

References:

[1] Arazy J., Contractive projections in C1 and C (1978)
[2] DOI: 10.1215/S0012-7094-92-06505-7 · Zbl 0779.47016 · doi:10.1215/S0012-7094-92-06505-7
[3] Cobos F., ”On the structure of bounded trilinear forms” (1997)
[4] DOI: 10.1112/S0024610799007504 · Zbl 0935.47022 · doi:10.1112/S0024610799007504
[5] Cobos F., Studia Math. 138 (1) pp 81– (2000)
[6] Cobos F., Rev. Real Acad. Ciencias Fisicas, Exactas y Naturales de Madrid 94 pp 441– (2000)
[7] DOI: 10.1007/978-0-8176-4771-1 · doi:10.1007/978-0-8176-4771-1
[8] DOI: 10.1090/S0002-9947-1989-0986027-X · doi:10.1090/S0002-9947-1989-0986027-X
[9] DOI: 10.1112/S0024610700001290 · Zbl 1031.42010 · doi:10.1112/S0024610700001290
[10] Pietsch A., Eigenvalues and s-numbers (1987) · Zbl 0615.47019
[11] Schur I., Vorlesungen über Invariantentheorie (1968) · Zbl 0159.03703 · doi:10.1007/978-3-642-95032-2
[12] Weyl H., The classical groups: their invariants and representations (1939) · JFM 65.0058.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.