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

$\langle A x,y\rangle=\langle x,A^{*}y\rangle$ for all $x,y \in \mathcal{H}$.

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

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

As such,

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

where $A^{T}$ is the transpose of $A$ and $\bar{A}$ 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, $A^{\dagger}$ is exclusively used for the adjoint while $A^{*}$ is interpreted as the same thing as $\bar{A}$.