On complementary subspaces of Hilbert space

Every pair {M,N} of non-trivial topologically complementary subspaces of a Hilbert space is unitarily equivalent to a pair of the form {G(-A) + K, G(A) + (0)} on a Hilbert space H + H + K. Here K is possibly (0), A is an element of B(H) is a positive injective contraction and G(+/-A) denotes the graph of +/-A. For such a pair {M, N} the following are equivalent: (i) {M, N} is similar to a pair in generic position; (ii) M and N have a common algebraic complement; (iii) {M, N} is similar to {G(X), G(Y)} for some operators X, Y on a Hilbert space. These conditions need not be satisfied. A second example is given (the first due to T. Kato), involving only compact operators, of a double triangle subspace lattice which is not similar to any operator double triangle.