Example of a positiv operator which is neither injective nor has a closed range. Let $E$ be a complex Hilbert space, $\mathcal{L}(E)$ be the algebra of all bounded linear operators on $E$.

I hope to get an example of $S\in \mathcal{L}(E)^+$ such that $S$ is neither injective nor has a closed range.

And thank you!

Hint: Take $E = \ell^2$ and choose the multiplication operator
$$(S x)_n = a_n \, x_n$$
for a sequence $(a_n)_{n \in \mathbb N}$.
Now, you should figure out which properties of $(a_n)$ ensure that


*

*$S$ is bounded

*$S$ is positive

*$S$ is not injective

*$S$ has not a closed range

