I know that the space of the all 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.