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^* represents 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 typically defined on a Hilbert space, which is a complete inner product space. Self-adjoint operators play a crucial role in various applications, including the study of quantum states and observable quantities in quantum physics.