The Banach space $C[0,1]$ is universal for all separable Banach spaces in the sense that for a separable Banach space $X$ there is an isometric isomorphism from $X$ to $C[0,1]$. My question is whether there is a universal operator in the following sense:

Do there exist separable spaces $X$ and $Y$ and a bounded linear operator $U:X \to Y$ so that if $A:E\to F$ ($E,F$ are separable Banach spaces) then $A=BUC$ where $C:E\to X$ and $B:Y\to F$. My guess is that the answer is no.

The reason I think the answer is no is that Johnson and Szankowski proved two results that somewhat contradict the affirmative: (1) There is no separable Banach space that is complementably universal for all separable Banach spaces (Studia 1976). (2) There is no separable Banach space that every compact operator factors through (JFA 2009).

If the answer is no to the first question then I'm interested in the same question for compact $A$.

The Banach space $C[0,1]$ is universal for all separable Banach spaces in the sense that for a separable Banach space $X$ there is an isometric isomorphism from $X$ into $C[0,1]$. My question is whether there is a universal operator in the following sense:

Do there exist separable spaces $X$ and $Y$ and a bounded linear operator $U:X \to Y$ so that if $A:E\to F$ ($E,F$ are separable Banach spaces) then $A=BUC$ where $C:E\to X$ and $B:Y\to F$. My guess is that the answer is no.

The reason I think the answer is no is that Johnson and Szankowski proved two results that somewhat contradict the affirmative: (1) There is no separable Banach space that is complementably universal for all separable Banach spaces (Studia 1976). (2) There is no separable Banach space that every compact operator factors through (JFA 2009).

If the answer is no to the first question then I'm interested in the same question for compact $A$.