Compact self-adjoint orthogonal operators

Question

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.

Answer 1

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 $XY=0$.

Answer 2

EDIT. This counterexample cannot be self-adjoint (see comment). Therefore this answer is wrong.

It is false, consider $x=y$ and let $x$ be any operator such that $x^2\ne 0$ and $x^4=0$ (you can make an explicit $4\times 4$ matrix example).