I would like to better understand the interplay between bounded operators on $L^2$ and operators that are given by distributional Schwartz kernels.
As someone who is usually more interested in $L^2$ and Sobolev spaces, I'm aware that many useful bounded operators on $L^2$ can be represented by their Schwartz kernels. However, it isn't clear to me what the relation between these two sets of operators is, perhaps because (it seems to me) discussions of distribution theory usually take place away from the Hilbert space setting.
In particular, I haven't been able to find explicit answers to the following questions:
Question 1: Can a bounded operator $T:L^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$ always be represented by a distributional Schwartz kernel?
Question 2: If not, is there an easy example of $T$ that cannot be represented in this way?
I would also appreciate any suggestions for references that address these or similarly minded concerns.
Answer to question 1: yes.
Indeed let $\iota_1:\mathcal{S}(\mathbb{R})\hookrightarrow L^2(\mathbb{R})$ and $\iota_2:L^2(\mathbb{R})\hookrightarrow \mathcal{S}'(\mathbb{R})$ be the canonical injection maps. Here $\mathcal{S}(\mathbb{R})$ is the Schwartz space of rapidly decaying smooth functions equipped with its natural Fréchet topology. Of course $L^2(\mathbb{R})$ is equipped with its usual Hilbert space topology. Finally, $\mathcal{S}'(\mathbb{R})$ is the space of temperate distributions equipped with its natural topology, i.e., the strong topology (not the weak-$\ast$ topology). Then with these choices of topologies, the linear maps $\iota_1,\iota_2$ are also continuous in the usual point set topology sense (not the lazy definition with sequences). Since $T$ is a bounded operator and therefore a continuous linear map $L^2(\mathbb{R})\rightarrow L^2(\mathbb{R})$, we have that $\iota_2\circ T\circ\iota_1:\mathcal{S}(\mathbb{R})\rightarrow\mathcal{S}'(\mathbb{R})$ is also linear continuous and therefore has a distributional kernel, by the Schwartz Kernel Theorem for the $\mathcal{S},\mathcal{S}'$ case.
More precisely, there is a unique distribution $\phi(x,y)$ in $\mathcal{S}_{x,y}'(\mathbb{R}^2)$ such that for all test function $f(y)$ in $\mathcal{S}_y(\mathbb{R})$ and $g(x)$ in $\mathcal{S}_x(\mathbb{R})$, $$ \langle\ g,T(f)\ \rangle_{L^2(\mathbb{R})}= \langle\ \langle \phi(x,y),f(y)\rangle_y\ ,g(x)\ \rangle_x =\langle\ \phi(x,y),\ g(x)f(y)\ \rangle_{x,y}\ . $$ The first bracket is the $L^2$ inner product while the following ones are distributional pairings which could heuristically be written as an integral with respect to some variable(s). Here the latter instead are indicated as subscripts. The first inequality is the one affirmed by the Kernel Theorem, whereas the second equality is automatic by Fubini's theorem for temperate distributions.