Self-adjoint operator

In mathematics, a self-adjoint operator is a densely defined linear operator mapping a complex Hilbert space onto itself and which is invariant under the unary operation of taking the adjoint. That is, if $A$ is an operator with a domain $H_{0}$ which is a dense subspace of a complex Hilbert space $H$ then it is self-adjoint if $A=A^{*}$, where $^{*}$ denotes the adjoint. Note that the adjoint of any densely defined linear operator is always well-defined and two operators A and B are said to be equal if they have a common domain and their values coincide on that domain.

On an infinite dimensional Hilbert space, a self-adjoint operator can be thought of as the analogy of a real symmetric matrix (i.e., a matrix which is its own transpose) or a Hermitian matrix in (i.e., a matrix which is its own Hermitian transpose) when these matrices are viewed as (bounded) linear operators on $\mathbb {R} ^{n}$ and $\mathbb {C} ^{n}$, respectively.

References

1. K. Yosida, Functional Analysis (6 ed.), ser. Classics in Mathematics, Berlin, Heidelberg, New York: Springer-Verlag, 1980