On Von Neumann's Problem Of A Semi-bounded Operator With Defect Index (1, 1)

For a semi-bounded operator with defect index (1, 1 ), we solve von Neumann's problem in a explicit way based on Krein's formula and rank one perturbation theory of a positive selfadjoint operator, only assuming the Friedrichs extension is obtained. In the last theorem, we point out that for every operator in this class, the image of the solution set of von Neumann' s problem under a natural bijection from theselfadjoint extension set to the unit circle S1 is in unit circle.