Spectral representations of unbounded non-linear operators on Hilbert space
Let \({\mathcal K}\) be a separable complex \(\infty\)-dimensional Hilbert space and let \({\mathcal F}\) be the Fock space of symmetric tensors over \({\mathcal K}\). We consider non-linear operators T from \({\mathcal K}\) to \({\mathcal F}\) defined on a dense subspace \({\mathcal D}\) in \({\mathcal K}\) with range in \({\mathcal F}\). A symmetry and reality condition is imposed on the operators T under consideration. They are generally unbounded and have different extensions \(\tilde T\) defined on subspaces \(\tilde {\mathcal D}\) in \({\mathcal K}\) containing \({\mathcal D}\). Generalizing a result of Arveson for bounded operators (alias functions from \({\mathcal K}\) to \({\mathcal F})\), we show that if T is affiliated with a maximal abelian von Neumann algebra in \(B({\mathcal K})\), then it follows that there is an extension\~T of T which is unitarily equivalent to a (non-linear) multiplication operator.
scientific article; zbMATH DE number 3861851