An elementary proof of the spectral theorem for unbounded operators
Bagby, Richard Julian
Master of Arts
One of the proofs of the spectral theorem for bounded operators begins by proving that a bounded, positive definite self-adjoint operator on a Hilbert space has a unique positive definite self-adjoint square root. From this result, I will show directly that an unbounded positive definite selfadjoint operator also has a unique square root. From this, I will derive the spectral theorem for unbounded self-adjoint operators. With this approach, the necessary results follow directly from elementary properties of operators on a Hilbert space. The resolution of the identity corresponding to an operator is obtained directly from the operator, rather than from the spectral resolution of related operators.