Consider the heat equation
$$ u_{t}=u_{xx},~~~~~u_0(x)=u(0,x)$$
with $u\colon [0,T]\times\mathbb{R}\to\mathbb{R}, (t,x)\mapsto u(t,x)$
and the evolution operator $E(T)$ with $E(T)u_0=u(T,x)$.
1.) Show that $u_0\star G_t$ with $G_t(x)=\frac{1}{\sqrt{4\pi t}}\exp(-\frac{x^2}{4t})$ is a solution of the heath equation problem above.
2.) Show, that $E(T)\colon L^2([0,1])\to L^2([0,1])$ is a continuous and compact operator.
I already showed 1.) by solving the partial differential equation with Fourier transformation. So far so good.
But I do not come along with the compactness in 2.).
(I showed the continuity with the Young inequation using the convolution - is that right?)
But how can I show the compactness of that operator?
Can anybody please help me?
Greetings