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.