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

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 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, ${\displaystyle H.}$

Consider a linear map ${\displaystyle A:H_{1}\to H_{2}}$ between Hilbert spaces. Without taking care of any details, the adjoint operator is the (in most cases uniquely defined) linear operator ${\displaystyle A^{*}:H_{2}\to H_{1}}$ fulfilling

where ${\displaystyle \langle \cdot ,\cdot \rangle _{H_{i}}}$ is the inner product in the Hilbert space ${\displaystyle H_{i}}$, which is linear in the first coordinate and conjugate linear in the second coordinate. Note the special case where both Hilbert spaces are identical and ${\displaystyle A}$ is an operator on that Hilbert space.

When one trades the inner product for the dual pairing, one can define the adjoint, also called the transpose, of an operator ${\displaystyle A:E\to F}$, where ${\displaystyle E,F}$ are Banach spaces with corresponding norms ${\displaystyle \|\cdot \|_{E},\|\cdot \|_{F}}$. Here (again not considering any technicalities), its adjoint operator is defined as ${\displaystyle A^{*}:F^{*}\to E^{*}}$ with

i.e., ${\displaystyle \left(A^{*}f\right)(u)=f(Au)}$ for ${\displaystyle f\in F^{*},u\in E}$.