In a course I'm taking we defined compact operators as a linear mapping $H\rightarrow H$, where $H$ is a Hilbert space, that maps bounded sets to relative compact ones. The lecturer mentioned that the reason we defined it like this and not as $M\rightarrow H$ with $M\subseteq H$ is that every such compact operator could be extended from $M$ to $H$. In passing he mentioned that this had something to do with the Hahn-Banach theorem.
My questions are:
- What exactly had he meant, how can one construct an extension ? I can't see the connection with said theorem since it involves linear functionals and not maps.
- Is this extension unique ?
- Is the argument that any such operator defined on a subset can be extend to the whole space even adequate ? I'm thinking that there may be examples operators that are naturally defined only on a subset of $H$ and extending them to $H$ would be inelegant. Could you provide such an example for compact operators ?
(To see what I mean by such an example I'm going to give one for symmetric operators: the differential operator that maps a function from $L^2(\Omega)$ to its second derivative is naturally only defined in a subset of $L^2(\Omega)$, since second derivatives of $L^2$-functions do not necessarily have to lie in $L^2(\Omega)$ again - now it would be possible to extend this operator to $L^2(\Omega)$ by letting it be $0$ outside its natural domain, since that would preserve symmetry, but it would be inelegant.)