 Ellipticity of a bounded Hilbert-space operator

In a variational Hilbert space setting, a bounded operator $L$ is elliptic or uniformly coercive when $\mathrm{Re}⟨Lv,v⟩\ge \gamma \Vert v{\Vert }^2$ for a constant $\gamma >0$. On a real Hilbert space this condition depends on the symmetric part of $L$ and does not imply self-adjointness. It differs from the principal-symbol condition defining an elliptic differential operator.