 Self-adjoint operator

A densely defined operator $T$ on a Hilbert space is self-adjoint when it equals its adjoint operator, including equality of their operator domains:For a bounded linear operator defined on the whole Hilbert space, this is equivalent tofor 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.

In the context of linear algebra and functional analysis, a self-adjoint operator (also known as a self-adjoint matrix in finite dimensions) is a specific type of linear operator that has a particular property regarding its adjoint.