I have the following problem: I'm given a linear bounded operator $P\in L(L^2([a,b]))$, $a,b\in \mathbb{R}$ and I want to find a sequence of approximating linear bounded operators $(P_n)_{n\geq 1}$ satisfying the following conditions:

1. $P_n \to P$ in $L(L^2([a,b]))$ as $n\to \infty$ (i.e. in the norm topology);
2. $P_n(H^1_0([a,b])) \subset H^1_0([a,b])$ for every $n\geq 1$;
3. $P_n : H^1_0([a,b]) \to H^1_0([a,b])$ is bounded.

Here, $H^1_0([a,b])$ denotes the Sobolev space of order $1$ with Dirichlet boundary conditions.

Is it possible to find such an approximating sequence for a general operator $P\in L(L^2([a,b]))$?

Thank you very much in advance!

Best, Luke