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Extreme points of the complex binary trilinear ball. (English) Zbl 0956.46013

On a Hilbert space \(H\), a trilinear form \(T: H\times H\times H \mapsto \mathbb C\) has the norm \(\|T\|:= \sup \{|T(x,y,z)|: \|x\|=\|y\|=\|z\|=1\}\). In the case when \(H = \mathbb C^2\) the authors first show that \(\|T\|= 1\) if and only if \(T\) can be represented in the special shape \[ T(x,y,z) = x_1y_1z_1 +b_1 x_1y_2z_2 +b_2x_2y_1z_2 +b_3x_2y_2z_1 +cx_2y_2z_2 \] and the complex coefficients \(b_1, b_2, b_3, c\) satisfy the inequality \[ |b_1|^2 + |b_2|^2 + |b_3|^2 + |c|^2/2 +|X|\leq 1 \] where \(X:= 2b_1b_2b_3 + c^2/2\). They further show that \(T\) is an extreme point of the unit ball if and only if equality holds above and either \(X=0\) or \(|X-c^2/2|< |X|+ |c|^2/2\). This answers a problem of R. Grząślewicz and K. John [Arch. Math. 50, No. 3, 264-269 (1988; Zbl 0653.47024)] who solved the real case. It is pointed out that if \(T\) is regarded as a bilinear form whose coefficients depend on \(x\) then its discriminant is a \(2\)-homogeneous polynomial in \(x\) the determinant of which is \(2X\). Finally, the authors use the characterization to give precise bounds for the inequalities between the above norm and the Hilbert-Schmidt norm of \(T\).

MSC:

46B20 Geometry and structure of normed linear spaces
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
15A69 Multilinear algebra, tensor calculus

Citations:

Zbl 0653.47024
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