I know that the space of all the bounded linear maps between two Banach spaces, denoted by $L(X,Y)$, has a relationship with the projective tensor space of $X$ and $Y$, $$({X \widehat\otimes_{\pi} Y})^* = L(X,Y^*).$$ Is there any relationship between the space of all compacts operators between the two spaces, denoted by $K(X,Y)$, and the projective tensor space of $X$ and $Y$?

Thanks in advance.