Hello All,

I apologize if the following question is too elementary. Any suggestion is greatly appreciated. Thank you.

Let $n\geqslant 4$, $X$ be an $n$-dimensional inner product space over $\mathbb{R}$ and let $T:\Lambda^{2}(X)\rightarrow \Lambda^{2}(X)$ be a symmetric linear operator satisfying $$ \langle T(\omega),\omega\rangle\geqslant 0,\text{ for all }\omega\in \Lambda^{2}(X)\text{ with }\omega\wedge\omega=0. $$ Is it true that, for some $A\in \Lambda^{4}(X)$, $$ \langle T(\omega),\omega\rangle\geqslant 0,\text{ for all }\omega\in \Lambda^{2}(X)\text{ with }\langle A;\omega\wedge\omega\rangle=0? $$