The spectral theorem for unbounded self-adjoint operators and Nelson's theorem.

Abstract

In this thesis, we will introduce the notion of unbounded operators on a Hilbert space. We will discuss the definition of the adjoint of an operator, and what it means for an operator to be self-adjoint. After that, we will restrict ourselves to bounded operators and prove the Spectral Theorem for normal bounded operators. The notion of a spectral measure will be introduced as well. After that, we return to unbounded operators and we will consider the Cayley-transform. With that and the spectral theorem for normal bounded operators, we prove the spectral theorem for unbounded self-adjoint operators. Then we look into the situations that two unbounded self-adjoint operators commute on a common domain. We then prove a theorem of Nelson that gives some criteria which imply that the spectral measures of the two operators commute. And finally we will consider two examples of unbounded operators that play a role in Quantum Mechanics.