Timeline for Extension of weakly compact operators from $\ell_1$ into $c_0$

7 events

when toggle format	what		by	license	comment
Mar 14, 2012 at 16:15	comment	added	Joaquin M. Gutierrez		Thank you very much, Professor Johnson, for your beautiful answer. Joaquin
Mar 13, 2012 at 22:09	history	edited	Yemon Choi	CC BY-SA 3.0	clarification
Mar 13, 2012 at 22:08	comment	added	Yemon Choi		Good points, Bill. I (mis)understood the question as asking for any extension to $E$ which contains a continuous linear image of $\ell^1$. See Matt's comment above.
Mar 13, 2012 at 20:49	comment	added	Bill Johnson		In his Memoirs "Extension of compact operators", on page 19 Lindenstrauss remarks that the formal identity from $\ell_2$ into $c_0$ does not have an extension to any copy of $\ell_\infty$ containing $\ell_2$ isomorphically. More generally, since $\ell_\infty$ has the Dunford-Pettis property and every bounded linear operator from $\ell_\infty$ to $c_0$ is weakly compact, there are severe restrictions on operators into $c_0$ which can be extended to a containing copy of $\ell_\infty$.
Mar 13, 2012 at 20:06	comment	added	Bill Johnson		Anyway, the identity $i_{1,2}$ from $\ell_1$ to $\ell_2$ has an extension to any space containing $\ell_1$ isomorphically, because $i_{1,2}$ is $2$-summing (and thus factors through $\ell_\infty$), hence also the identity from $\ell_1$ to $c_0$ has an extension.
Mar 13, 2012 at 19:56	comment	added	Bill Johnson		Why would the extension give a projection, Yemon?
Mar 13, 2012 at 17:43	history	answered	Yemon Choi	CC BY-SA 3.0