Examples of self adjoint compact operators on Hilbert spaces As the titles says, I'm looking for examples of self adjoint compact operators on Hilbert spaces. 
So far I know of the diagonal operator on $\ell^2$,
$$
(Tx)_i = \alpha_ix_i 
$$
for some sequence $\alpha_i \to 0$;  and the Hilbert Schmidt integral operator in $L^2(\Omega)$,
$$
Kg = \int_{\Omega}K(x,y)g(y)dy
$$
with $K \in L^2(\Omega^2)$ a symmetric Hilbert-Schmidt kernel.
I would also like to know of some applications that use Hilbert Schmidt integral operators.
Edit: I'd be really grateful to know about the behaviour of the norm of the operators as well.
Thanks in advance.

An important example is the inverse of the Laplace operator: $T=(-\Delta)^{-1}$ as mapping from $H^1_0(\Omega)^*$ to $H^1_0(\Omega)$. Here, $\Omega$ is an open and bounded domain in $\mathbb R^n$. That is, given $z\in H^1_0(\Omega)$, $u:=Tz$ is the weak solution of 
$$
-\Delta u = z,
$$
which is equivalent to
$$
\int_\Omega \nabla u \cdot\nabla v \ dx = \int_\Omega zv \ dx \quad\forall v\in H^1_0(\Omega).
$$
