What is "spectral development" of a linear operator?

Question

I want to find the "spectral development" (or spectral series, spectral decomposition, not sure how to translate to english) of the linear operator.

What is that? I can't even google it. I think it is not singular value decomposition.

The operaror is $T(x_1, x_2,...) = (0, x_1, \frac{x_2}{2}, \frac{x_3}{3},...)$ from $\ell^2$ to $\ell^2$.

Answer 1

This is a compact operator with no eigenvalues: the spectrum consists of $0$ alone. So the decomposition is trivial: the generalized eigenspace for $0$ is the whole space.