Compact self-adjoint orthogonal operators Let $H$ be a Hilbert space and $x,y$ are self-adjoint compact linear operators acting in $H$. If $x^2\cdot y^2 =0$ then is it true that $x\cdot y=0$? Thanks.

If $X$ is a selfadjoint operator on a Hilbert space, then $X^2x=0$ iff $Xx=0$ because $X^2x=0\implies 0=\langle X^2x,x\rangle = \langle Xx,Xx\rangle=\|Xx\|^2$.

Therefore, if $X,Y$ are selfadjoint and $X^2Y^2=0$, it follows that $XY^2=0$. Taking adjoints gives $Y^2X=0$ and, hence, $YX=0$. Taking adjoints again gives the desired result that $XY=0$.