A Hermitian operator is a special type of linear operator in quantum mechanics that has important properties. It is defined by the condition that it is equal to its own adjoint, meaning that the inner product of two vectors remains unchanged when the operator is applied. This characteristic ensures that the eigenvalues of a Hermitian operator are real numbers, which is crucial for physical observables like position and momentum. In quantum mechanics, Hermitian operators are used to represent measurable quantities, such as energy and angular momentum. The eigenstates of a Hermitian operator form a complete basis, allowing any quantum state to be expressed as a combination of these states. This makes Hermitian operators fundamental in the mathematical framework of quantum theory, particularly in the context of quantum states and wave functions.