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Hermitian adjoints

Suppose $\mathscr{H}$ is a Hilbert space with an inner product $⟨\cdot ,\cdot ⟩$. Consider a continuous linear operator $A:\mathscr{H}\to \mathscr{H}$. One can show that there exists a unique continuous linear operator $A^{*}:\mathscr{H}\to \mathscr{H}$ with the following property:

$⟨Ax,y⟩=⟨x,A^{*}y⟩$ for all $x,y\in \mathscr{H}$.

This is a generalization of the concept of an adjoint matrix (also known as a conjugate transpose, Hermitian conjugate, or Hermitian adjoint). The adjoint of an $m\times n$ matrix $A$ with complex entries is the $n\times m$ matrix whose entries are defined by

$(A^{*}{)}_{\mathrm{ij}}=\overline{A_{\mathrm{ji}}}$.

As such,

$A^{*}=(\overline{A}{)}^T=\overline{A^T}$

where $A^T$ is the transpose of $A$ and $\overline{A}$ is the matrix with complex conjugate entries of $A$.

Hermitian matrices

A matrix, $A$, is said to be Hermitian if

$A^{*}=A$

where $A^{*}$ is the Hermitian adjoint of $A$.

Notation

The notation used here for the adjoint, $A^{*}$, is commonly used in linear algebraic circles (as is $A^H$). In quantum mechanics, $A^{†}$ is exclusively used for the adjoint while $A^{*}$ is interpreted as the same thing as $\overline{A}$.