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 $\left\{G(-A)\oplus K,G(A)\oplus(0)\right\}$ on a Hilbert space $H\oplus H\oplus K$. Here $K$ is possibly $(0)$, $A\in\mathcal{B}(H)$ is a positive injective contraction and $G(\pm A)$ denotes the graph of $\pm A$. 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 $\left\{G(X),G(Y)\right\}$ 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.