Let
and let
be a -finite measure space.
For , define on by .
I need to find the following:
, , and
where
is not invertible
: there is a sequence in X such that for all n and and
and
Your help is invaluable - I have no idea how to do this.
Let
and let
be a -finite measure space.
For , define on by .
I need to find the following:
, , and
where
is not invertible
: there is a sequence in X such that for all n and and
and
Your help is invaluable - I have no idea how to do this.
The point spectrum is the set of eigenvalues of . If is an eigenvalue, with eigenfunction f, then . So for almost all t in X. Thus almost everywhere. But f is an eigenfunction, so it cannot be 0 almost everywhere. Therefore there is a set of positive measure on which . Hence .
The approximate point spectrum is probably harder to describe. I don't offhand know what it is, but I would guess that it is the set of complex numbers such that for every . See if you can prove that!