Timeline for 2-summing vs Hilbert-Schmidt norm for extended operator between Hilbert and Banach space
Current License: CC BY-SA 4.0
12 events
| when toggle format | what | by | license | comment | |
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| Mar 16, 2021 at 22:10 | vote | accept | n_flanders | ||
| Mar 16, 2021 at 17:55 | comment | added | Dirk Werner | Thanks for your PS! | |
| Mar 16, 2021 at 17:55 | answer | added | Dirk Werner | timeline score: 1 | |
| Mar 16, 2021 at 12:01 | comment | added | n_flanders | PS: In case you happen to be the author of the famous German FA book, thanks for writing such a great text! I'm learning from it self-dependently and like it a lot. It is rigorous while also appealing at intuition, exactly as it should be. | |
| Mar 16, 2021 at 12:00 | comment | added | n_flanders | @DirkWerner You are absolutely right that I misread the proposition, my apologies for this. So just to confirm: In your modified example, $\pi_2(iT_n)=1$ while $\pi_2(T_n)$ is of order $\sqrt{n}$. As the latter norm explodes when $n \to \infty$, a bound of the form $\pi_2(iT_2) \leq \varphi( \pi_2(T) )$ as asked for in my modified question can't hold, correct? If so, please write this as an answer so that I can accept it. In any case, thanks again for your help! | |
| Mar 16, 2021 at 11:29 | comment | added | Dirk Werner | @nflanders I think my example, when normalised, fits your setup, with $b(x,y)= n^{-1/2}\sum_{k=1}^n x_k y_k$ so that $|b(x,y)|\le \|x\|_2\|y\|_\infty$ (i.e., $C=1$). So one should replace $T=T_n$ above with $T_n/\sqrt{n}$. -- I think you have misread Prop.\ 1.5.3 that does not say what you quote, but that $\pi_2(T)$ equals the (quasi-) norm of $T$ in the product operator ideal $\mathfrak{H}\circ \mathfrak{P}_2$, cf.\ ibidem D.1.10. | |
| Mar 16, 2021 at 4:57 | history | edited | n_flanders | CC BY-SA 4.0 |
added 47 characters in body
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| Mar 16, 2021 at 4:54 | comment | added | n_flanders | @DirkWerner Thanks for the comment, in general what I'm asking for can of course not hold, I was hoping that the special additional structure of $T_2$ would make a difference. Also, I should add that a non-linear dependence would be fine as well and have adjusted my question. In fact, in the very useful reference you're quoting there is Proposition 1.5.3, which says that if $T$ is 2-summing, $S$ is continuous and factors through a Hilbert space, then $T$ and $ST$ have the same 2-summing norm, which at least shows that $T_2$ has finite 2-summing norm. | |
| Mar 15, 2021 at 19:23 | comment | added | Dirk Werner | I think what you are asking is this: If $H$ is a Hilbert space, $X$ ($=B^*$) is a Banach space, $T:H\to X$ is an operator, $i: X\to H$ is a continuous injection, can one estimate the 2-summing norm $\pi_2(T)$ by a multiple of $\pi_2(iT)$? Take $H=\ell_2$, $X=\ell_1$, $i=$ the identity mapping and $T=T_n=$ the projection onto the first $n$ coordinates. Then $\pi_2(iT_n)= \sqrt{n}$, but $\pi_2(T_n)$ is of order $n$; see 1.6.8 in Pietsch's Eigenvalues and $s$-Numbers. | |
| Mar 15, 2021 at 10:19 | history | edited | n_flanders | CC BY-SA 4.0 |
removed nonsense
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| Mar 15, 2021 at 9:52 | review | First posts | |||
| Mar 15, 2021 at 10:00 | |||||
| Mar 15, 2021 at 9:52 | history | asked | n_flanders | CC BY-SA 4.0 |