Injectivity of $L2$ Operator with Delta Distribution in Frequency Domain

I'm trying to understand if an operator $T: L^2(\mathbb{R}) \to H$ is injective. The operator $T$ is well-defined in the sense that $H$ is defined as the image of $L^2(\mathbb{R})$ under $T$. We have a representation of $T$ in the frequency domain as a multiplication operator with the distribution $D$, which is a sum of two terms:

1. $m(\omega)$: a smooth function which is nonzero everywhere
2. A delta distribution (Dirac comb)

So, in the frequency domain, $T$ acts as:

$T(F) = D \cdot F = (m + \delta) \cdot F$

where $F$ is the Fourier transform of an $L^2(\mathbb{R})$ function, $m$ is the smooth function, and $\delta$ is the delta distribution.

The fact that $m$ is nonzero everywhere suggests that the operator $T$ might be injective, as it doesn't eliminate any frequency data and can always be inverted (by undoing the multiplication by $m$).

However, I'm struggling with how to handle the delta distribution part. If the Fourier transform of the input function were smooth, then this multiplication would make sense, but the Fourier transform of an $L^2$ function is itself in $L^2$, which may not be smooth enough for a straightforward multiplication with a delta distribution.

My questions are:

1. Can we use any density arguments, given that $C^\infty_c$ (smooth functions with compact support) is dense in $L^2$ and its Fourier transform is smooth, to infer injectivity for all of $L^2$?

2. Is it possible to use approximate sequences on the Fourier transform of the $L^2$ input function to handle the delta as a function or distributionally in a way that would allow us to infer injectivity?

3. More generally, how can we rigorously define and analyze the action of $T$ on $L^2$ functions, given the presence of the delta distribution in the frequency domain representation?

Any insights or suggestions on how to approach this problem would be greatly appreciated. I'm particularly interested in understanding if and how we can extend results from "nice" functions (where the multiplication with delta is well-defined) to all of $L^2(\mathbb{R})$.