Consider the operator $M$ on $\ell^2(\mathbb{Z})$ defined by for $u\in \ell^2(\mathbb Z)$

$$Mu(n)=\frac{1}{\vert n \vert+1}u(n).$$ This is a compact operator!

Then, let $l$ be the left-shift and $r$ the right-shift on $\ell^2(\mathbb Z).$

We consider the compact operator on $\ell^2(\mathbb Z;\mathbb C^2)$ defined by

$$T:=\begin{pmatrix} 0 & l M \\ rM & 0 \end{pmatrix}$$

My question is: Even though $T$ is not normal, since $$T^*T= \begin{pmatrix} MlrM & 0 \\ 0 & Mr lM \end{pmatrix}=M^2$$ whereas $$TT^*= \begin{pmatrix} lM^2 r & 0 \\ 0 & rM^2 l \end{pmatrix}\neq M^2$$ does $T$ have eigenvalues?