= Self-adjoint operator
{wiki}
= Self-adjointness
{synonym}
= Self-adjoint
{synonym}
A <densely defined operator> $T$ on a <Hilbert space> is self-adjoint when it equals its <adjoint operator>, including equality of their <operator domains>:
$$
D(T)=D(T^*),\qquad Tv=T^*v\quad(v\in D(T)).
$$
For a <bounded linear operator> defined on the whole <Hilbert space>, this is equivalent to
$$
\langle Tv,w\rangle=\langle v,Tw\rangle
$$
for all vectors $v,w$. For an <unbounded self-adjoint operator>, the domain condition is essential: the inner-product identity on $D(T)$ alone gives only a <symmetric operator>.
Back to article page
