Let $H$ be a Hilbert space and let $T$ be defined on $H\oplus H$ \begin{equation} T=\left(\begin{array}{cc} 0 & A \\ B & 0 \\ \end{array} \right).\end{equation}
What do we know wbout the spectrum of $T$ in terms of that of either $A$ or $B$ or both? I could say that in case $AB=BA$ that $\sigma(T^2)=\sigma(AB)$. Any other remark or reference or related result is most appreciated.
Many thanks.
Math
First, $0\in \sigma(T)$ if and only if $0\in\sigma(A) \cup \sigma(B)$.
Second, for $\lambda\ne 0$, the operator $\lambda I-T$ is continuously invertible if and only if $\lambda I - AB$ is continuously invertible if and only if $\lambda I - BA$ is continuously invertible.
This can be proven by considering the system $(\lambda I-T)x=b$: solve one of the equations and plug this solution into the other.
Hence we get: $$ \sigma(T) = ( (\sigma(A)\cup\sigma(B)) \cap \{0\} ) \cup (\sigma(AB)\setminus\{0\}) $$