Operator blocks and quadruples of subspaces: Classification and the eigenvalue completion problem

A direct connection is made between finite dimensional operator blocks and a certain class of quadruples of subspaces. It is shown that the problem of classifying operator blocks under block similarity is equivalent to the problem of classifying the corresponding quadruples of subspaces under isomorphism of subspace quadruples. This leads in a natural way to a problem of extension for quadruples of subspaces, namely the analogue of the eigenvalue completion problem for operator blocks.