Essential Self-Adjointness of Linear Operators on Hilbert Spaces and Spectral Theory

Abstract

Essential Self-Adjointness of Linear Operators on Hilbert Spaces and Spectral Theory Abstract: Unbounded linear operators are ubiquitous in mathematics and its applications. For example, differential operators are not in general bounded. The need for a framework in which to study these operators naturally leads to unbounded opera- tor theory. There is a spectral theorem for unbounded operators, allowing a functional calculus to be constructed for any self-adjoint operator. Self-adjointness is essential for this theorem, so questions of when an operator is merely symmetric, or self-adjoint, or perhaps essentially self-adjoint, are of interest. In this paper, we prove a version of the spectral theorem for tuples of unbounded operators, which is somewhat of a folklore theorem in the literature. We then apply this spectral theorem to generalise a result concerning the essential self-adjointness of a differential operator on Euclidean space satisfying certain conditions. The proof in this paper generalises the result to an arbitrary Hilbert space, where the operators are not assumed to be differential operators.