On complementary subspaces of Hilbert space

Longstaff, W.; Panaia, Oreste

doi:10.1090/S0002-9939-98-04547-X

On complementary subspaces of Hilbert space
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by W. E. Longstaff and Oreste Panaia PDF

Proc. Amer. Math. Soc. 126 (1998), 3019-3026 Request permission

Abstract:

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.

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