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.

Printable View

- March 23rd 2009, 10:25 AMNuscSpectrum of Linear Operator
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. - March 24th 2009, 02:05 PMOpalg
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!