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How do I find the norm of the following operator i.e. how to find $\lVert T_z\rVert$ and $\lVert l\rVert$?

1) Let $z\in \ell^\infty$ and $T_z\colon \ell^p\to\ell^p$ with $(T_zx)(n)=z(n)\cdot x(n).$ What my thoughts were to use Banach-Steinhaus theorem but it seems straight forward and I don't know if I am right.

$\lVert T_z\rVert _p \leqslant\lVert z\lVert \cdot n\cdot\lVert x\rVert_p n=n^2\lVert x\rVert _p$ so if I choose $x=1$ then I get $\lVert T_z\rVert =n^2$.

2) Let $0\leqslant t_1\leqslant\cdots\leqslant t_n=1$ and $\alpha_1,\dots,\alpha_n \in K$ , $l\colon C([0,1])\to K$ with $l(x)=\sum_{i=1}^n \alpha_i x(t_i)$.

How to I find operator norm in this case as well? I am quite sure I am not right. I would be glad if I could get some help. Definitely some hints would be great! Thanks in advance.

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    Well, "finding the operator norm" mean you have to show $\|T\|=K$ for some $K$, so you have to show $\geq$ and $\leq$.2012-11-13

1 Answers 1

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  1. As for each $n$, $|z(n)x(n)|^p\leqslant \lVert a\rVert_{\infty}|x(n)|^p$, then we certainly have $\lVert T_z\rVert\geqslant \lVert a\rVert_{\infty}$. To get the other inequality, fix $\delta$ and pick $k$ such that $|a(k)|\geqslant \lVert a\rVert_{\infty}-\delta$ (the case $a=0$ is obvious).

  2. We assume $t_j$ distinct. Let $f_j$ a continuous map such that $f_j(t_j)=e^{i\theta_j}$, where $e^{i\theta_j}\alpha_j=|\alpha_j|$ and $f_j(t_k)=0$ if $k\neq j$. We can choose the $f_j$'s such that $\lVert \sum_{j=1}^nf_j\rVert=1$.

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    $||a||_{\infty}$ is $sup z(n)$ right ? and after that you have made a claim that certainly $||T_z||\ge ||a||_{\infty}$, that means $||x(n)||\ge 1$ but that i don't see. How can i agrue that i can pick $k$ in the next line , is it because if there was no such k which was $a(k)\ge ||a||_{\infty}-\delta$ then it would contradict our assumption right ? And the second one i am not able to understand your construction and how would it answer the question . Thanks a lot for helping .2012-11-13