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

Abstract

We characterize the sets of norm one vectors x1, ..., xk in a Hilbert space H such that there exists a k-linear symmetric form attaining its norm at (x1, ..., xk). We prove that in the bilinear case, any two vectors satisfy this property. However, for k ≥ 3 only collinear vectors satisfy this property in the complex case, while in the real case this is equivalent to x1, ..., xk 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 Ls(kH).