Essential Adjoints: Spectral Decomposition and Symmetric Operators

This lecture discusses the importance of closed operators in achieving a spectral decomposition of self-adjoint operators, focusing on the case of multiplication operators on Hilbert spaces. It covers the relationship between closed operators and their adjoints, proving key propositions and lemmas. The lecture also introduces the concept of essential self-adjointness, highlighting conditions for an operator to be essentially self-adjoint. The Von Neumann theorem is presented, establishing criteria for an operator to be self-adjoint. Examples of symmetric operators, such as T₁ and -A₁, are analyzed to illustrate the theory discussed. The lecture concludes by demonstrating the symmetry and properties of these operators.