Vol. 111, No. 1, 1984

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Spectral representations of unbounded nonlinear operators on Hilbert space

Palle E. T. Jorgensen

Vol. 111 (1984), No. 1, 93–104

Let be a separable complex -dimensional Hilbert space and let be the Fock space of symmetric tensors over . We consider non-linear operators T from to defined on a dense subspace 𝒟 in with range in . 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 containing 𝒟. Generalizing a result of Arveson for bounded operators (alias functions from to ), we show that if T is affiliated with a maximal abelian von Neumann algebra in B(), then it follows that there is an extension T of T which is unitarily equivalent to a (non-linear) multiplication operator.

Mathematical Subject Classification 2000
Primary: 47H99
Secondary: 46L99, 47B25
Received: 23 February 1982
Published: 1 March 1984
Palle E. T. Jorgensen