A quantitative version of the Johnson–Rosenthal theorem

Abstract

Let $X,Y$ be Banach spaces. We define $${\alpha }_Y\left(X\right)=sup\hspace{0.17em}\left\{\right|T^{-1}{|}^{-1}:T:Y\to X\text{is an isomorphism with}\left|T\right|\le 1\}.$$ If there is no isomorphism from $Y$ to $X$, we set ${\alpha }_Y\left(X\right)=0$, and

$${\gamma }_Y\left(X\right)=sup\hspace{0.17em}\left\{\delta \right(T):T:X\to Y\text{is a surjective operator with}|T|\le 1\},$$ where $\delta \left(T\right)=sup\hspace{0.17em}\{\delta >0:\delta B_Y\subseteq T{B}_X\}$. If there is no surjective operator from $X$ onto $Y$, we set ${\gamma }_Y\left(X\right)=0$. We prove that for a separable space $X$, ${\alpha }_{l_1}\left(X^{\ast }\right)={\gamma }_{c_0}\left(X\right)$ and ${\alpha }_{L_1}\left(X^{\ast }\right)={\gamma }_{C(\Delta )}\left(X\right)={\gamma }_{C[0,1]}\left(X\right)$.