Coercive operator

A bounded operator $T$ on a Hilbert space is coercive when $\mathrm{Re}⟨x,Tx⟩\ge c\Vert x{\Vert }^2$ for some $c>0$. This estimate implies injectivity and bounds an inverse on the range by $1/c$.

For a map $A:M\to X^{\ast }$ on a subset of a real Banach space, coercivity with respect to $u_0\in M$ means the displayed limit as $u\in M$ and $\Vert u-u_0\Vert \to \infty$. Here the brackets denote the dual pairing. This growth condition is useful in existence theorems for monotone operators. A linear Hilbert-space estimate $⟨Au,u⟩\ge c\Vert u{\Vert }^2$, $c>0$, implies the condition at zero. Monotonicity alone does not: the zero operator is monotone and fails this growth condition on an unbounded space.

On l2 sequence space ${\ell }^2$, let $(Lu{)}_n=u_n/n$ and $f_n=1/n$. This bounded self-adjoint operator has $⟨Lu,u⟩>0$ for nonzero $u$, but the equation $Lu=f$ requires $u_n=1$, which is not in ${\ell }^2$. Taking the first $N$ coordinates equal to one makes $I(u)=-{\sum }_{n=1}^{N}1/n\to -\infty$. Thus strict positivity in isolation does not give existence in that Hilbert space; a coercive bilinear form and a continuous forcing functional in its norm are essential hypotheses for the usual minimization proof.

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.