Let $1<p<2$. My question is: Is any operator from $l_{p}$ into Tsirelson's space $T$ compact?
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1$\begingroup$ Yes. Same argument that works into $\ell_1$. $\endgroup$– Bill JohnsonNov 22, 2015 at 0:12
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$\begingroup$ @BillJohnson, of course if by the Tsirelson space Dongyang understands your version, namely the dual of the original construction. In the case of the original Tsierlson space, non-compact operators exist. $\endgroup$– Tomasz KaniaNov 22, 2015 at 0:17
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$\begingroup$ Is my question true for $p\geq 2$? $\endgroup$– Dongyang ChenNov 22, 2015 at 0:26
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$\begingroup$ @TomekKania: In the case of the original Tsierlson space, non-compact operators exist. Could you tell me the relevant reference? Thank you. $\endgroup$– Dongyang ChenNov 22, 2015 at 0:51
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2$\begingroup$ @DongyangChen, If I understand your question and the earlier comments correctly then maybe the paper Remarks on some basic properties of Tsirelson's space by Castillo and Sanchez may be of interest to you; the paper can be found at siba-ese.unisalento.it/index.php/notemat/article/view/1304 $\endgroup$– Philip BrookerNov 22, 2015 at 4:00
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