Jtdylktuy

Let $Tin ell^1$, $Tx = (lambda_1x_1,dots,lambda_nx_n,dots)$. Want to show that if $T$ is compact, then $lambda_nto0$.

I know for $pin(1,infty]$, canonical basis $e_n rightharpoonup 0$ (so $T(e_n)rightharpoonup 0$), and $T(e_n)in overline{T(B(0,1))}$, which is compact, so we can get $T(e_n)to 0$, and $lambda_nto 0$.

But what should I do with this problem when $p=1$, there is no weak convergence to $0$. Can someone help me with this? Thanks

Suppose that $Tcolonell^1 to ell^1$ is compact. Then the sequence $left(Te_nright)_{ngeqslant 1}$ admits a subsequence $left(Te_{n_k}right)_{kgeqslant 1}$ which converges to some $v$ (strongly) in $ell^1$. Look at $leftlVert Te_{n_{k+1}}-Te_{n_k}rightrVert_1$ to conclude that $lambda_{n_k}to 0$.

Now apply the previous result to $left(Te_{N_j}right)_{jgeqslant 1}$ for a fixed sequence $N_juparrow infty$ instead of $left(Te_nright)_{ngeqslant 1}$ to see that each subsequence of $left(lambda_nright)_{ngeqslant 1}$ admit a further subsequence with converges to $0$.