I was wondering whether there is a general methodology of going about showing whether bounded, linear operators are open mappings or closed operators? Let's say I meet a space and an operator such as the weighted right-shift.
$T : ℓ^ 1 (\mathbb{N}) → ℓ^ 1 (\mathbb{N}), T(x_1, x_2, x_3 . . .) = (x_2, x_3/ 2 , x_4 /(2^ 2) , . . .).$
Is there some fixed way of doing this kind of problem? I know that you have to use the open mapping and the closed graph theorem, but it is not clear to me, for example, how to show that the domain of $T$ is a closed subset of $ℓ^1$