Spectral Decomposition of Unbounded Operators

This lecture covers the spectral decomposition of non-bounded operators, defining normal and self-adjoint operators in the non-bounded case, establishing the existence of unique elements in the domain of operators, and proving the properties of the operator 1 + A*A. The lecture concludes with the spectral theorem for self-adjoint non-bounded operators, presenting a countable family of Borel measures and a unitary operator U that diagonalizes the operator A on a specific domain.