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

摘要

We characterize the sets of norm one vectors x(1),, x(k) in a Hilbert space 74 such that there exists a k-linear symmetric form attaining its norm at (x(1),..., x(k)). 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 xi,, 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,C8(H). (C) 2018 Elsevier Inc. All rights reserved.