Stuck on computing operator norm

Question

I'm very new to the practice of computing operator norm and for every question I seem to get stuck unless it is trivially easy (or probability even when it is still trivially easy). Basically I cannot seem to be able to compute even the simplest operator norm. For example, consider the linear operator

$$ T: \ell^1 \rightarrow \ell^{\infty}: (x_k)_{k=1}^{\infty} \mapsto (x_k/2^{k})_{k=1}^{\infty}.$$ In order to compute the operator norm I can see that $$||Tx||_{\ell^{\infty}} = \sup_{k}|x_k/2^k| \leq \sum_{k=1}^{\infty}|x_k|/2^k \leq \sum_{k=1}^{\infty}|x_k|=||x||_{\ell^1}.$$ Then by taking sup over all $ \||x||_{\ell^1} = 1 $ we have $||T||\leq 1$. So a good guess seems to be that $||T||=1$. But I cannot prove the other inequality. For right now the most reasonable approach I can think of is to cook up a sequence such that $ ||x||_{\ell^1} \leq 1 $ and $||Tx||_{\ell^\infty} = 1$. Can anyone help me with this problem?

In fact, after going through some questions (without solving anyof them) it seem I can always easily obtain some upper bound of $||T||$ but then get stuck on the other inequality as soon as the function $T$ becomes reasonably complex (or not). Is this even the right mind set when attacking such a problem? I would like some general advice if anyone could be kind enough to offer them to me.

Thank you!

Answer 1

In order to have a satisfactory lower bound for the norm of an operator on $\ell^p$ space, we evaluate the operator at the vectors of the "canonical basis" (the $e_n$ where all the coordinates of $e_n$ are zero except the $n$-th, which is one) or linear combinations of such vectors. For example, we are sure that $$ \lVert T\rVert \geq \sup_{n\geqslant 1}\lVert Te_n\rVert_{\ell^\infty}=\sup_{n\geqslant 1}2^{-n}\lVert e_n\rVert_{\ell^\infty}=1/2. $$ At this step, we cannot conclude for the value of the norm. However, since $$\sup_{k\geqslant 1}|x_k/2^k|\leqslant \frac 12\sup_{k\geqslant 1}|x_k|\leqslant \frac 12\lVert x\rVert_{\ell^1},$$ it turns out that the norm is $1/2$.

Answer 2

Your approach is good: First try to find an upper bound for $\Vert Tx \Vert /\Vert x \Vert$, then look for a point $x_0$ at which the maximum of $\Vert Tx \Vert /\Vert x \Vert$ is attained (not always possible) or a sequence $(x_n)$ such that $\Vert Tx_n \Vert /\Vert x_n \Vert \rightarrow \Vert T \Vert$.

As in the example you have mentioned, an upper bound usually comes with a chain of inequalities. At this point you can check each inequality separately and ask yourself whether there is an $x_0$ for which the inequality is actually an equality or if there is a sequence $(x_n)$ for which the inequality becomes closer and closer to an equality. In the example at hand you have two inequalities: $$ \sup_{n\ge 1} \vert x_n/2^n\vert \le \sum_{n\ge 1} \vert x_n \vert /2^n $$ and $$ \sum_{n\ge 1} \vert x_n \vert /2^n \le \sum_{n\ge 1} \vert x_n\vert. $$ Hint: The second inequality can be improved. Once this is done you can find an $x_0$ for which both inequalities are actually equalities. What is this $x_0$?