Multilinear Forms of Hilbert Type and Some Other Distinguished Forms

Abstract.

We give some new examples of bounded multilinear forms on the Hilbert spaces ℓ₂ and L₂ (0, ∞). We characterize those which are compact or Hilbert-Schmidt. In particular, we study m-linear forms (m ≥ 3) on ℓ₂ which can be regarded as the multilinear analogue of the famous Hilbert matrix. We also determine the norm of the permanent on $\mathbb{K}^{n} ,$ where $\mathbb{K} = \mathbb{R}\;{\text{or}}\;\mathbb{C}.$