# Norm of powers of the Volterra Operator

• Jun 17th 2011, 12:27 PM
Insensus
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!.

$\displaystyle$V: C[0,1] \rightarrow C[0,1]$\\(Vf)(s) = \displaystyle{\int_0^s f(t)dt}$
Show $\displaystyle$\left|\left|V^n\right|\right| \leq \frac{1}{n!}$$Now I know how the operator norm is defined \displaystyle \left|\left|T\right|\right| = \displaystyle{\sup_{\left|\left|x\right|\right|=1} } \left|\left|Tx\right|\right| and I've been throwing it at this problem for at least 3 hours, but I can't seem to crack it. I've so far assumed that induction would be the path to choose but I'm starting to doubt that, yet I see no other method. I hope someone is able to nudge me in the right direction. Thanks, Insensus • Jun 17th 2011, 12:47 PM Also sprach Zarathustra Re: Norm of powers of the Volterra Operator Quote: Originally Posted by Insensus Hey there, For a course in Functional Analysis we were asked to show that the norm of the n-th power of the Volterra operator is less than or equal to 1 over n!. \displaystyle V: C[0,1] \rightarrow C[0,1]\\(Vf)(s) = \displaystyle{\int_0^s f(t)dt} Show \displaystyle \left|\left|V^n\right|\right| \leq \frac{1}{n!}$$

Now I know how the operator norm is defined
$\displaystyle$\left|\left|T\right|\right| = \displaystyle{\sup_{\left|\left|x\right|\right|=1} } \left|\left|Tx\right|\right|$and I've been throwing it at this problem for at least 3 hours, but I can't seem to crack it. I've so far assumed that induction would be the path to choose but I'm starting to doubt that, yet I see no other method. I hope someone is able to nudge me in the right direction. Thanks, Insensus Maybe this will help you: SpringerLink - Integral Equations and Operator Theory, Volume 27, Number 4 Hope that is free... • Jun 17th 2011, 03:04 PM Jose27 Re: Norm of powers of the Volterra Operator How about this: Consider$\displaystyle f_n(t)=t^n$for$\displaystyle n\geq 0$then it's not difficult to see that$\displaystyle V^k(f_n)(t) = \frac{t^{n+k}}{(n+k)...(n+1)}t^n$from which we immediately deduce that$\displaystyle \| V^k(p)(t)\| \leq \frac{1}{k!}$where$\displaystyle p\$ is any polynomial. Now use Weierstrass' approximation theorem.
• Jun 18th 2011, 02:49 AM
Insensus
Re: Norm of powers of the Volterra Operator
That's very nice, thanks.