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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

On complementary subspaces of Hilbert space


Authors: W. E. Longstaff and Oreste Panaia
Journal: Proc. Amer. Math. Soc. 126 (1998), 3019-3026
MSC (1991): Primary 46C05
DOI: https://doi.org/10.1090/S0002-9939-98-04547-X
MathSciNet review: 1468197
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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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Additional Information

W. E. Longstaff
Affiliation: Department of Mathematics, The University of Western Australia, Nedlands, Western Australia 6907, Australia
Email: longstaff@maths.uwa.edu.au

Oreste Panaia
Affiliation: Department of Mathematics, The University of Western Australia, Nedlands, Western Australia 6907, Australia
Email: oreste@maths.uwa.edu.au

DOI: https://doi.org/10.1090/S0002-9939-98-04547-X
Received by editor(s): March 14, 1997
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society