On complementary subspaces of Hilbert space

Abstract

Every pair { M , N } \{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 ) } \left \{G(-A)\oplus K,G(A)\oplus (0)\right \} on a Hilbert space H ⊕ H ⊕ K H\oplus H\oplus K . Here K K is possibly ( 0 ) (0) , A ∈ B ( H ) A\in \mathcal {B}(H) is a positive injective contraction and G ( ± A ) G(\pm A) denotes the graph of ± A \pm A . For such a pair { M , N } \{M,N\} the following are equivalent: (i) { M , N } \{M,N\} is similar to a pair in generic position; (ii) M M and N N have a common algebraic complement; (iii) { M , N } \{M,N\} is similar to { G ( X ) , G ( Y ) } \left \{G(X),G(Y)\right \} for some operators X , Y 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.