Potential operator

A mapping $ A $ of a Banach space $ X $ into the dual space $ X ^ {*} $ that is the gradient of some functional $ f \in X ^ {*} $, i.e. is such that

$$ \langle A x , h \rangle = \lim\limits _ { t\rightarrow } 0 \ \frac{f ( x + t h ) - f ( x) }{t} . $$

For instance, any bounded self-adjoint operator $ A $ defined on a Hilbert space $ H $ is potential:

$$ Ax = \mathop{\rm grad} \left \{ \frac{1}{2} \langle A x , x \rangle \right \} ,\ \ x \in H . $$

References

[1]	M.M. Vainberg, "Variational method and method of monotone operators in the theory of nonlinear equations" , Wiley (1973) (Translated from Russian)
[2]	H. Gajewski, K. Gröger, K. Zacharias, "Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen" , Akademie Verlag (1974)