Let $v$ and $w$ be an $n \times 1$ and $m \times 1$ unit norm vectors, respectively. Also, let $H$ an $n \times m$ matrix. We denote by vectors $a^*$ and $a^T$ the conjugate transpose and transpose of vector $a$, respectively.
The author of a paper uses the following equality: $v^* H w= h^* (w \otimes v^*)$, where $\otimes$ is the Kronecker product and $h$ (of dimension $mn \times 1$) is the vectorization of matrix $H$.
I think their claim is incorrect and the write relation is $v^* H w= h^T (w \otimes (v^*)^T)$.
Could you please confirm if my result is correct? and if not, could you give me the right relation ?
The initial expression is linear in $H$, hence we can not use $h^*$, only $h^T$ and $h$.
After that, the initial expression is antilinear in $v$, hence we can only use $v^*$ and $(v^*)^T$.
Similarly, we can use only $w$ and $w^T$. The above arguments support your claim about conjugates.
Then again, $h^T$ is a $1\times mn$ matrix (vector), then so must be the matrix $w\otimes (v^*)$, which is false (the latter object is a $m\times n$ matrix).
Therefore, the conclusion is that your variant formula is the correct one.