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I have one unsolved problem related to the ellipticity of some linear differential operator.

In this question, I guess, I oversimplified the assumptions and the answer was not as I suspected.

So let $X^*$ mean the conjugate transpose of matrix $X.$

Assume that we have two matrices $H,V$ such that $(H^2=0\wedge V^2=0\wedge VH=HV).$

Question. Does $VH^*H=H^*VH$ then?

I hope this holds. If not, I am close to doomed.

PS. I added homological-algebra tag cause I guess some trics associated with assumption $H^2 = 0\wedge V^2 = 0$ migth be needed.

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    Taking the conjugate transpose on both sides and rearranging, we can rephrase the question as whether $$ H^*(HV^* - V^*H) = 0 $$ which looks nicer to me, somehow. – 2017-01-31

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It seems that you are close to doomed after all. For example, consider $$ H=V=\pmatrix{0&1\\0&0} $$

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    Yes, it works. I am little bit angry and I honestly thank you very much. – 2017-01-31
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    I would be grateful if you could give me some hints with non-simplified verison of this problem which I posted here: http://math.stackexchange.com/questions/2122908/hx-vy-wedge-vx-hz-wedge-va-x-hb-wedge-h2-0-wedge-v2-0-wedge-vh-hv-i – 2017-01-31