Consider a densely defined, self-adjoint operator
$$
H: \mathcal{D} \rightarrow \mathscr{H}.
$$
Assume for simplicity that $H$ is nonnegative.
We want to effectively restrict this operator $H$ to a subspace $P\mathscr{H}$. Here $P: \mathscr{H} \rightarrow \mathscr{H}$ is an orthogonal projection. We assume that the subspace $P\mathscr{H} $ to which we want to restrict is not contained in the domain of $H$ (i.e. $P\mathscr{H} \nsubseteq \mathcal{D}$).
If $P$ is finite dimensional, we may define a regularized operator $H_f$ acting on $P\mathscr{H}$ as follows:
Consider a continuous injective function $ f: [0,\infty) \rightarrow [0,\infty) $ satisfying
$$ \lim_{\lambda \rightarrow \infty} f(\lambda) = 0. $$
The operator $f(H): \mathscr{H} \rightarrow \mathscr{H}$ is thus bounded. Hence the operator $$ Pf(H)P: P\mathscr{H} \rightarrow P\mathscr{H} $$
is well-defined. It is not hard to see that it is also non-negative. Thus we may define an effective operator $H_f: P\mathscr{H} \rightarrow P\mathscr{H}$ as $$ H_f = f^{-1}(Pf(H)P). $$
- Are there references that discuss these ideas?
- Is there a neat characterization of $H_f$. (Even if only for the resolvent case $f(\cdot) = (1 + \cdot)^{-1}$? )
- Is anything known in the setting where $P$ is not finite?
