Inspired by Pitt's theorem and this post we ask the following question:
First we remind the Pitt's theorem and also introduce a particular Banach space as averaging of $\ell^p$ spaces for $1\leq p \leq 2$
Pitt's theorem: Every bounded operator $T:\ell^q \to \ell^p$, $p<q$ is a compact operator.
Now consider $X=\ell_1$ with new norm $|x|=\int_1^2 |x|_pdp$ the completion of this normed space is denoted by $\tilde{X}$.
Question: Assume that $T:\tilde{X}\to \ell^1$ is a bounded operator. Is $T$ necessarilly a compact operator?
