I'm reading some material about single-variable distribution theory. More specifically, I was checking some theorems of the convolution algebra $\mathcal{D}_+$, where $\mathcal{D}_+$ is the space of all distributions with support bounded on the left. The author defined the convolution over $\mathcal{D}_+$ and then presented Titchmarsh's theorem stating that the convolution algebra $\mathcal{D}_+$ has no zero divisors, and thus if an inverse of an element of $\mathcal{D}_+$ exists it must be unique. In what follows he introduces an example of solving a linear differential equation, and here is my question:
He defines the ordinary differential operator $D := \frac{d^n}{dt^n} + a_{n-1} \frac{d^{n-1}}{dt^{n-1}} + \dots + a_0$. And then he says: Lets define the inverse of $D\delta$, where $\delta$ is the Dirac distribution.
What does the symbol $D\delta$ refer to? What kind of object is this? From the context it implies that this is an element of $\mathcal{D}_+$, since he considers a linear differential equation of the form $D\delta \ast x = Dx = 0$. How should one understand $D\delta$?
And what about $x$ here, is it considered to be a distribution too? Can you provide a calculations using the definition of $D\delta$ to show that $D\delta \ast x = Dx$