Suppose I have the following matrix. $$A ``=" \left[\begin{matrix} a & b\\ L & d\end{matrix}\right]$$ where $L$ is a functional on $l^2$. I want to find $A^*$, but it seems like the usual thing, $A^* = (\overline{A})^T$, fails because of $L$. I want to check if $A$ is self-adjoint, so suppose $L$ is self-adjoint and $a, b, d$ are real, how do I compute the adjoint of $A$?
Thank you.