Let ℋ be a separable complex
∞-dimensional Hilbert space and let ℱ be the Fock space of symmetric tensors over
ℋ. We consider non-linear operators T from ℋ to ℱ defined on a dense subspace 𝒟 in
ℋ with range in ℱ. A symmetry and reality condition is imposed on the operators T
under consideration. They are generally unbounded and have different extensions T
defined on subspaces 𝒟 in ℋ containing 𝒟. Generalizing a result of Arveson for
bounded operators (alias functions from ℋ to ℱ), we show that if T is affiliated with
a maximal abelian von Neumann algebra in B(ℋ), then it follows that there is an
extension T of T which is unitarily equivalent to a (non-linear) multiplication
operator.
