# Anti-commutative hermitian operators

I have some trouble to prove the next statements:

1. Let $$A,B$$ two anti-commutative hermitian operators, i.e. $$\{A,B\}=AB+BA=0$$. Does $$A$$ and $$B$$ share any eigenket?.

2. If $$U$$ is an unitary operator such that $$U=A+iB$$ with $$A$$ and $$B$$ hermitian operators with non-degenerated eigenvalues. Show that any eigenvector of $$A$$ is an eigenvector of $$B$$.

For $$1.$$ "Let $$v$$ a common eigenket, thus $$Av=av$$ and $$Bv=bv$$ so:

$$0=(AB+BA)v=ABv+BAv=(ab+ba)v$$, since $$A,B$$ are hermitian and $$a,b$$ are real, and $$ab=-ba=-ab$$ thus $$ab=0$$ but I don't know if this tells me something, and for $$2.$$ I don't know how to start, Any help?

• According to your derivation for (1), it said if $A$ and $B$ share a common eigenket, then $a = 0$ or $b = 0$. Equivalent, it said if both $a$ and $b$ are non-zero, then $A$ and $B$ does not share any eigenket – K_inverse Sep 24 '18 at 7:17