Take the 2-minute tour ×  Regarding this question,  Why do
spectral projections give norm approximations?  I have figured
out what is meant by spectral projection, and have thus found
the answer as well. A spectral projection is the image of $x$
under a step/indicator function defined on its spectrum, which
is hence an orthogonal projection on some closed subspace. (Here
I am using the Bounded Borel functional calculus.)  Okay, it's a
projection. And I said the word "spectrum" in the above
paragraph. But surely it's more "spectral" than that. Does the
image of the so-obtained projection have anything to do with $x$
of a spectral nature? (This is an open-ended question, so please
qualify what sort of spectral significance there is. But some
specific interpretations are below):    1. Suppose $\lambda$ is
an eigenvalue of $x$ which hence means that its in the spectrum
of $x$. Does the indicator function of the one point $\lambda$
applied to x via the Borel Functional Calculus project onto the
eigenspace associat