Jacobian of operator Facts
We know Jacobian of a multivariate multivalued function $f(x)$ is simply a matrix contains the partial derivative: $\partial f_i/ \partial x_j$
Extension to operator?
But what if we replace $f$ to an operator, $\mathcal{L}: L_2 \mapsto L_2$, which is mapping function to function? There is a similar concept of Jacobian?
 A: The appropriate generalization you are looking for is the Frechet derivative.
Which is the generalization of the Jacobian to mappings between arbitrary Banach spaces. Although it should be noted that the concept of the derivative could be generalized to arbitrary topological vector spaces since you need a vector space structure to take difference quotients, and you need a topology to define limits.
A: You should not say, that the Jacobian is a matrix containing the partial derivatives of the components. Consider the function:
$$f\colon\Bbb R^2\to\Bbb R,(x,y)\mapsto\left\{\begin{array}[cc]
& 1 & ;y=x^2\;\text{and}\;(x,y)\neq(0,0) \\
0 & ;\text{otherwise} 
\end{array}\right..$$
Both the partial derivatives $\partial_xf(0,0)=0$ and $\partial_yf(0,0)=0$ exist, but $f$ is not even continuous in $(0,0)$ and therefore also the Jacobian doesn't exist.
I don't know if a similar concept exists for mapping functions to functions, but there is for mapping functions to scalars, also called functionals. It is called the functional derivative (see here, here or here) and for example of great use in quantum field theory.
