showing something odd for an operator

Hi,

How do I do the following:

Define for . Show that for all the following holds:

.

(The hint given is that one should compute the resolvent of V)

So far I have been able to show that the given operator is compact and has no eigenvalues. This then gives me the spectrum of , . From this, since the spectrum is the complement of the resolvent, i get that the . What now?

Many Thanks!

Re: showing something odd for an operator

Quote:

Originally Posted by

**FalexF** Hi,

How do I do the following:

Define

for

.

Are you sure about that? In that definition, is independent of x. I think it is more likely that it should be In that case, V is the Volterra integral operator. You could try looking online (or better still, in a library) for information about it.

Re: showing something odd for an operator

Oh yes! It's supposed to go from 0 to x. And I actually (by accident) sumbled upon the real name (Volterra) a couple of hours ago. So, I've been googeling like a mad man since then. And I've worked with it before (that's why I knew that it's compact) but I've had no luck when trying to solve the above. Any ideas?

Re: showing something odd for an operator

Quote:

Originally Posted by

**FalexF** Hi,

.

Now, I'm thinking that this is obvious. You see I have a thm that says: .

Now if i apply that to the above I should get the follwoing:

This will now tend to infinity as long as our fixed . Correct? Hmmm... I am however not using the fact that ...

Re: showing something odd for an operator

Quote:

Originally Posted by

**FalexF** Now, I'm thinking that this is obvious. You see I have a thm that says:

.

Now if I apply that to the above I should get the following:

This will now tend to infinity as long as our fixed

. Correct? Hmmm... I am however not using the fact that

...

No, it will not tend to infinity, because when is very small, will be large, and so will be *very* small.

It looks to me as though you should follow the hint in the question and compute the resolvent Do this in a totally nonrigorous way using methods of Differential Equations 101.

If then . That is, Assume that f and g are both differentiable. Also assume (for convenience later on) that f(0) = 0. Differentiate, to get Solve that differential equation by the Diff. Eq. 101 method of introducing an integrating factor so as to write the equation in the form Integrate, getting

(the lower limit 0 for the integral ensures that f(0) = 0, which is why I made that assumption earlier).

I reckon that is what the hint wanted you to do. Why is that useful? Well, we want to show that can be very large when is small. So we want to find f and g as in the boxed equation such that is very much larger than I'm not sure quite how to achieve that. The best I can do is to try taking The -norm of g is then given by But and so If you then work out , it comes to something of the order of which looks promising. But I don't see where that in the problem comes from.