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What is the norm of integral operators $A$ in $L_2(0,1)$?

$Ax(t)=\int_0^tx(s)ds$

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    Where is $B$?${}{}$2012-06-09
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    Sorry, it was a mistake, there's only one operator, $A$.2012-06-09
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    Then you should edit your question :)2012-06-09
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    $\left\|A\right\|_{2}\le 1$. This can be shown using some integral inequalities and Holder's inequality. I have not been able to show that this bound is tight, unfortunately.2012-06-09
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    I suspect it might be easier to look at $A$ in terms of the basis $e_n(t) = e^{i n 2\pi t}$.2012-06-09

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It's enough to use Schwarz inequality in the following manner:

$$ \| A x \|^2 = \int_0^1 \left| \int_0^t x(s) \, ds \right|^2 dt = \int_0^1 \left| \int_0^t \sqrt{\cos \frac{\pi}{2}s} \cdot \frac{x(s)}{\sqrt{\cos \frac{\pi}{2}s}} \,ds \right|^2 dt \le \int_0^1 \left( \int_0^t \cos \frac{\pi}{2}s \, ds \int_0^t \frac{|x(s)|^2}{\cos \frac{\pi}{2}s}\right) dt = \frac{2}{\pi} \int_0^1 \int_0^t \sin \frac{\pi}{2}t \, \frac{|x(s)|^2}{\cos \frac{\pi}{2}s} \, ds\,dt = \frac{2}{\pi}\int_0^1 \left( \int_s^1 \sin \frac{\pi}{2} t \, dt \right) \frac{|x(s)|^2}{\cos \frac{\pi}{2}s} \,ds = \left( \frac{2}{\pi} \right)^2 \| x \|^2 $$
Equality holds for $x(s) = \cos \frac{\pi}{2}s$.

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    This is reverse engineering :) When you know the answer, it is always much easier to get it.2012-06-09
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    @Norbert :D it's also quite common exercise ;)2012-06-09
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    @qoqosz It should be $x(s)=\sqrt2\cos\frac\pi 2s$. I am only pointing this out because I spent the past hour trying to figure out why I wasn't getting the right result :)2016-05-28
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    How is it that you are able to obtain that $|\int_0^t \sqrt{\cos(\frac{\pi s}{2})} \frac{x(s)}{\sqrt{\cos(\frac{\pi s}{2})}}|^2 \leq \int_0^t \cos(\frac{\pi s}{2})ds \int_0^t \frac{|x(s)|^2}{\cos(\frac{\pi s}{2})}ds$? Is this always true or why can you do this in this case?2017-11-19
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The norm of the Volterra operator is $2/\pi$. I will try to recall the proof; the bound suggests that the optimum occurs for some trigonometric polynomial, say $\cos(\pi x/2)$.

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It is Problem 188 in the book by P. Halmos, "A Hilbert space problem book". In the solution, the author writes that "A direct approach seems to lead nowhere." The norm is indeed $2/\pi$, and is computed through the adjoint $A^*$ and a suitable kernel. It is a rather long proof, so please try to read it on Halmos' book.

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    If the book is not handy, I also recently wrote up the proof here: http://math.stackexchange.com/questions/151425/spectrum-of-indefinite-integral-operators/151444#1514442012-06-10