Can both of these linear operators be unbounded?

Question

Suppose $A$, $B$, $C$, and $D$ are compact and injective linear operators in $L_2[0,1]$ and that $AB^{*}=CD^{*}$. Is it possible that both $D^{-1}B$ and $C^{-1}A$ are unbounded?

Answer 1

Yes, that's possible. Choose an ONB $(e_n)_{n\geq 1}$ of $L^2([0,1])$ and let \begin{align*} Ae_n=\begin{cases}n^{-1}e_n&\text{if }n\text{ even},\\ n^{-2}e_n&\text{if }n\text{ odd},\end{cases}\\ Be_n=\begin{cases}n^{-2}e_n&\text{if }n\text{ even},\\ n^{-1}e_n&\text{if }n\text{ odd}.\end{cases} \end{align*} Both are self-adjoint compact injective operators and they commute. With $C=B$ and $D=A$ we get $AB=CD$ and $$ D^{-1}Be_n=ne_n $$ for $n$ odd and $$ C^{-1}Ae_n=ne_n $$ for $n$ even.

Hence both $D^{-1}B$ and $C^{-1}A$ are unbounded.