On orthogonal multiplication in complex Hilbert spaces. (English. Russian original) Zbl 1003.46014

Vakhaniya, N. N.; Kandelaki, N. P.

Theory Probab. Math. Stat. 63, 27-32 (2001); translation from Teor. Jmovirn. Mat. Stat. 63, 26-31 (2000).

Let \(H\) be a Hilbert space. A bilinear mapping \(p:H\times H\to H\) satisfying the condition \(\|p(x,y)\|=\|x\|\cdot\|y\|\) for all \(x,y\in H\) is called orthogonal multiplication in \(H\). The orthogonal multiplication is called normalized if for this multiplication there exists an one-side unit. The authors prove that in the complex infinite-dimensional separable Hilbert space \(H\) there exists an orthogonal multiplication and does not exists any normalized orthogonal multiplication.

Reviewer: A.D.Borisenko (Kyïv)

MSC:

46C05	Hilbert and pre-Hilbert spaces: geometry and topology (including spaces with semidefinite inner product)
60G15	Gaussian processes

Keywords:

orthogonal multiplication; complex Hilbert space