I know that the set of the all linear applications between two spaces, denoted by L(X,Y) have a relationship with the tensor proyective space of X and Y, $$(\hat{X \oplus_{\pi} Y})^* = L(X,Y^*)$$. There is any relationship between the set of the all compacts operators between two spaces, denoted by K(X,Y) and the tensor proyective space of X and Y.

Thanks in advance.