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

TL;DR

This paper characterizes the vectors in a Hilbert space where symmetric multilinear forms attain their maximum norm, revealing differences between real and complex cases and providing applications to tensor products and the geometry of operator spaces.

Contribution

It provides a complete characterization of vectors where symmetric multilinear forms attain their norm, highlighting distinctions between real and complex Hilbert spaces.

Findings

In the bilinear case, any two vectors satisfy the norm attainment property.

For $k extgreater 2$, only collinear vectors do in the complex case.

In the real case, vectors span a subspace of dimension at most 2.

Abstract

We characterize the sets of norm one vectors $\mathbf{x}_1,\ldots,\mathbf{x}_k$ in a Hilbert space $\mathcal H$ such that there exists a $k$-linear symmetric form attaining its norm at $(\textbf{x}_1,\ldots,\mathbf{x}_k)$. 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 $\mathbf{x}_1,\ldots,\mathbf{x}_k$ 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 $\mathcal L_s(^k\mathcal{H})$.