Norm of powers of the Volterra Operator

Hey there,

For a course in Functional Analysis we were asked to show that the norm of the n-th power of the Volterra operator is less than or equal to 1 over n!.

Show

Now I know how the operator norm is defined

and I've been throwing it at this problem for at least 3 hours, but I can't seem to crack it.

I've so far assumed that induction would be the path to choose but I'm starting to doubt that, yet I see no other method.

I hope someone is able to nudge me in the right direction.

Thanks,

Insensus

Re: Norm of powers of the Volterra Operator

Quote:

Originally Posted by

**Insensus** Hey there,

For a course in Functional Analysis we were asked to show that the norm of the n-th power of the Volterra operator is less than or equal to 1 over n!.

Show

Now I know how the operator norm is defined

and I've been throwing it at this problem for at least 3 hours, but I can't seem to crack it.

I've so far assumed that induction would be the path to choose but I'm starting to doubt that, yet I see no other method.

I hope someone is able to nudge me in the right direction.

Thanks,

Insensus

Maybe this will help you:

SpringerLink - Integral Equations and Operator Theory, Volume 27, Number 4

Hope that is free...

Re: Norm of powers of the Volterra Operator

How about this:

Consider for then it's not difficult to see that from which we immediately deduce that where is any polynomial. Now use Weierstrass' approximation theorem.

Re: Norm of powers of the Volterra Operator

That's very nice, thanks.