A Hermitian operator, also called a symmetric operator, is a densely defined linear operator 
on a Hilbert space, with inner
product conjugate-linear in its first argument, satisfying
 |
for all 
in the domain of 
. Equivalently, 
is contained in its adjoint
operator 
.
A self-adjoint operator additionally requires equality
of the two domains. Thus every self-adjoint
operator is Hermitian, but a Hermitian unbounded operator need not be self-adjoint.
Some physics literature uses the two terms synonymously.
For a bounded operator defined on the entire Hilbert space, Hermitian and self-adjoint are equivalent. In finite dimensions and with respect to an orthonormal basis, a Hermitian operator is represented by a Hermitian matrix.
If 
and 
,
the Hermitian identity gives
 |
Taking 
shows that every eigenvalue is real,
and eigenvectors belonging to distinct eigenvalues
are orthogonal. Stronger conclusions such as
an orthonormal basis of eigenvectors
belong to the spectral theorem and require its
hypotheses. For a differential operator,
the formal adjoint, operator domain, and boundary conditions
must all be distinguished.
