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.References
- P. A. Fillmore and J. P. Williams, On operator ranges, Advances in Math. 7 (1971), 254–281. MR 293441, DOI 10.1016/S0001-8708(71)80006-3
- P. R. Halmos, Two subspaces, Trans. Amer. Math. Soc. 144 (1969), 381–389. MR 251519, DOI 10.1090/S0002-9947-1969-0251519-5
- P. R. Halmos, Reflexive lattices of subspaces, J. London Math. Soc. (2) 4 (1971), 257–263. MR 288612, DOI 10.1112/jlms/s2-4.2.257
- Paul R. Halmos, A Hilbert space problem book, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto, Ont.-London, 1967. MR 0208368
- Edward Kissin, On some reflexive lattices of subspaces, J. Operator Theory 25 (1991), no. 1, 141–162. MR 1191257
- M. S. Lambrou and W. E. Longstaff, Finite rank operators leaving double triangles invariant, J. London Math. Soc. (2) 45 (1992), no. 1, 153–168. MR 1157558, DOI 10.1112/jlms/s2-45.1.153
- W. E. Longstaff, Nonreflexive double triangles, J. Austral. Math. Soc. Ser. A 35 (1983), no. 3, 349–356. MR 712812, DOI 10.1017/S144678870002704X
- H. K. Middleton, On the reflexivity and transitvity of non-distributive subspace lattices, Ph.D. Thesis, University of Western Australia, 1988.
- Manos Papadakis, On isomorphisms between certain non-CSL algebras, Proc. Amer. Math. Soc. 119 (1993), no. 4, 1157–1164. MR 1235101, DOI 10.1090/S0002-9939-1993-1235101-8
- Heydar Radjavi and Peter Rosenthal, Invariant subspaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 77, Springer-Verlag, New York-Heidelberg, 1973. MR 0367682, DOI 10.1007/978-3-642-65574-6
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
- Received by editor(s): March 14, 1997
- Communicated by: Palle E. T. Jorgensen
- © Copyright 1998 American Mathematical Society
- 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