A Hermitian operator is a special type of linear operator used in quantum mechanics and mathematics. It has the property 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 represent measurable quantities, such as energy or angular momentum. The eigenstates of these operators correspond to possible outcomes of measurements, and the probabilities of these outcomes can be calculated using the associated eigenvalues. This makes Hermitian operators fundamental to the mathematical framework of quantum theory.