Let $X= [x_{i,j}]\in M_{n}(\mathbb{C})$ and its transpose is defined by $$ X^t= [x_{j,i}]. $$
Then the linear operators $X, X^t: \mathbb{C}^n\to \mathbb{C}^n$ is defined by $$ X\begin{bmatrix} h_{1}\\ \vdots\\ h_{n} \end{bmatrix}= \begin{bmatrix} \sum_{k=1}^{n}x_{1k}h_{k}\\ \vdots\\ \sum_{k=1}^{n}x_{nk}h_{k}\\ \end{bmatrix} \text{ for all }\begin{bmatrix} h_{1}\\ \vdots\\ h_{n} \end{bmatrix}\in \mathbb{C}^n,$$
and $$ X^t\begin{bmatrix} h_{1}\\ \vdots\\ h_{n} \end{bmatrix}= \begin{bmatrix} \sum_{k=1}^{n}x_{ki}h_{k}\\ \vdots\\ \sum_{k=1}^{n}x_{kn}h_{k}\\ \end{bmatrix} \text{ for all }\begin{bmatrix} h_{1}\\ \vdots\\ h_{n} \end{bmatrix}\in \mathbb{C}^n. $$
Can yo say the operator norm of $X$ and $X^t$ are equal ( ie. $\Vert X\Vert = \Vert X^t\Vert$)?
I am curious to know when such equality happens? Thanks in advance for providing the suggestions.
Caution: We have not consider here $X^t$ as a conjugate transpose. In such case the result is very well known.
\begin{align} \left\|X\right\| &= \max_{\left\|h\right\|=1} \left\|Xh\right\|\\ &= \max_{\left\|h\right\|=1} \left\|\overline X^T h\right\|\\ &= \max_{\left\|\overline h\right\|=1} \left\|\overline X^T \overline h\right\|\\ &= \max_{\left\|h\right\|=1} \left\|\overline {X^T h}\right\|\\ &= \max_{\left\|h\right\|=1} \left\|{X^T h}\right\| = \left\|X^T\right\|\\ \end{align}