Two remarks concerning the equation $\Pi_p(X,\cdot)=I_p(X,\cdot)$

It is proved that the analog of Grothendieck's theorem is valid for a disk-algebra “up to a logarithmic factor”. Namely, if $T\in\mathscr L(C_A,L^1)$ and $\operatorname{rank}T\le n$ then $\pi_2(t)\le C(1+\log n)\|T\|$. The question of whether the logarithmic factor is actually necessary remains open. It is also established that $C^*_A$ is a space of cotype $q$ for any $q$, $q>2$. The proofs are based on a theorem of Mityagin–Pelchinskii: $\pi_p(T)\le c\cdot p\cdot i_p(T)$, $p\ge2$, for any operator $T$ acting from a disk-algebra to an arbitrary Banach space.