Let $X = \ell^2$ The operators $\textbf{L}$ and $\textbf{R}$ are defined as
$\textbf{R}x = (0, a_0, a_1...) \;\; \textbf{L}x = (a_1, a_2, a_3...) $
show that they are the transposes of one another $\textbf{L}' = \textbf{R},\;\; \textbf{R}' = \textbf{L}. $