There is no bounded linear surjection between $\ell_p$ spaces

Question

For $1\leq p,q<\infty$, $p\ne q$, how to prove that there is no bounded linear operator $T:\ell_p\to \ell_q$ such that $T$ is surjective?

I've tried to use Pitt's theorem, but without success.

Answer 1

By assumption $p\neq q$.

Case 1: $p,q\in(1,+\infty)$. Assume we have a surjection $T:\ell_p\to\ell_q$, then $T^*:\ell_{q'}\to \ell_{p'}$ is an embedding. Since $p',q'\in(1,+\infty)$, we get a contradiction because by corollary of Pitt's theorem theses spaces are totally incomparable. Thus for this case a desired surjection doesn't exists.

Case 2: $q=1, p\in(1,+\infty)$. Similarly, we have an embedding $T^*:\ell_\infty\to\ell_{p'}$. Since $p'\in(1,+\infty)$, then $\ell_{p'}$ is separable, but $\ell_\infty$ is not. So $T^*$ can't be an embedding. For this case we don't have a surjection either.

Case 3: $p=1, q\in(1,+\infty)$. Since $q\in(1,+\infty)$, then $\ell_q$ is separble. It is remains to recall that any separable Banach space is quotient of $\ell_1$. So this is the only case when a surjection does exists.