Let $H=(H,\langle\cdot ,\cdot\rangle)$ be a Hilbert space and $\mathcal{L}:D(\mathcal{L}) \subset H \times H \longrightarrow H \times H$ a linear operator. Consider his quadratic form $\mathcal{Q}$ associated to $\mathcal{L}$ given by $$\mathcal{Q}(f,g)=\langle\mathcal{L}(f,g),(f,g)\rangle,\;\forall \; f,g \in D(\mathcal{L}).$$
If there exists $(f,g) \in D(\mathcal{L})$ such that $\mathcal{Q}(f,g)<0$. Can I conclude that $\mathcal {L} $ takes on a negative eigenvalue? This is in general true?