Hermitian adjoint

In mathematics, specifically in operator theory, each linear operator ${\displaystyle A}$ on an inner product space defines a Hermitian adjoint (or adjoint) operator ${\displaystyle A^{*}}$ on that space according to the rule

${\displaystyle \langle Ax,y\rangle =\langle x,A^{*}y\rangle ,}$

where ${\displaystyle \langle \cdot ,\cdot \rangle }$ is the inner product on the vector space.

The adjoint may also be called the Hermitian conjugate or simply the Hermitian^[1] after Charles Hermite. It is often denoted by A^† in fields like physics, especially when used in conjunction with bra–ket notation in quantum mechanics. In finite dimensions where operators can be represented by matrices, the Hermitian adjoint is given by the conjugate transpose (also known as the Hermitian transpose).

The above definition of an adjoint operator extends verbatim to bounded linear operators on Hilbert spaces ${\displaystyle H}$. The definition has been further extended to include unbounded densely defined operators, whose domain is topologically dense in, but not necessarily equal to, ${\displaystyle H.}$