  It is proved that, for each pair (m,n) of non-negative integers, there is a
Banach space X for which the group K_0(B(X)) is isomorphic to m copies of the
integers and the group K_1(B(X)) is isomorphic to n copies of the integers.
Along the way we compute the K-groups of all closed ideals of operators
contained in the ideal of strictly singular operators, and we derive some
results about the existence of splittings of certain short exact sequences.
