A self-adjoint operator is a type of linear operator that is equal to its own adjoint. In mathematical terms, if A is a self-adjoint operator, then A = A^* , where A^* denotes the adjoint of A . Self-adjoint operators are important in quantum mechanics and functional analysis because they guarantee real eigenvalues and orthogonal eigenvectors. These operators are often represented in Hilbert spaces, which are complete inner product spaces. Self-adjoint operators play a crucial role in the spectral theorem, which states that any self-adjoint operator can be diagonalized by an orthonormal basis of eigenvectors. This property is essential for understanding physical systems in quantum mechanics.