Symmetric multilinear forms on Hilbert spaces: where do they attain their norm?

Abstract

We characterize the sets of norm one vectors ${\mathbf{x}}_1,\dots ,{\mathbf{x}}_k$ in a Hilbert space $\mathcal{H}$ such that there exists a $k$-linear symmetric form attaining its norm at $({\text{x}}_1,\dots ,{\mathbf{x}}_k)$. We prove that in the bilinear case, any two vectors satisfy this property. However, for $k\ge 3$ only collinear vectors satisfy this property in the complex case, while in the real case this is equivalent to ${\mathbf{x}}_1,\dots ,{\mathbf{x}}_k$ spanning a subspace of dimension at most 2. We use these results to obtain some applications to symmetric multilinear forms, symmetric tensor products and the exposed points of the unit ball of ${\mathcal{L}}_s{(}^k\mathcal{H})$.

17 pages