Abstract
A family of subspaces of a complex separable Hilbert space is transitive if every bounded operator which leaves each of its members invariant is scalar. This article surveys some results concerning transitive families of small cardinality, and adds some new ones. The minimum cardinality of a transitive family in finite dimensions (greater than 2) is 4. In infinite dimensions a transitive pair of linear manifolds exists but the minimum cardinality of a transitive family of dense operator ranges or norm-closed subspaces is not known. However, a transitive family of dense operator ranges with 5 elements can be found, and so can a transitive family of norm-closed subspaces with 4 elements. In finite dimensions (> 1) three nest algebras (corresponding to maximal nests) can intersect in the scalar operators, but two cannot. It is not known if this is the case in infinite dimensions for maximal nests of type ω + 1. Four such nest algebras can intersect in the scalar operators.
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References
Halmos, P. R.: Ten problems in Hilbert space. Bull. Amer. Math. Soc., 76, 887–933 (1970)
Lambrou, M. S., Longstaff, W. E.: Finite rank operators leaving double triangles invariant. J. London Math. Soc., 45(2), 153–168 (1992)
Lambrou, M. S., Longstaff, W. E.: Small transitive families of subspaces in finite-dimensions. Linear Alg. & Applic., 357, 229–245 (2002)
Hadwin, D. W., Longstaff, W. E., Peter R.: Small transitive lattices. Proc. Amer. Math. Soc., 87(1), 121–124 (1983)
Harrison, K. J., Radjavi, H., Rosenthal, P.: A transitive medial subspace lattice. Proc. Amer. Math. Soc., 28, 119–121 (1971)
Fillmore, P. A., Williams, J. P.: On operator ranges. Advances in Math., 7, 254–281 (1971)
Longstaff, W. E.: Small transitive families of dense operator ranges. Int. Eq. & Operator Th., 45(3), 343–350 (2003)
Foiaş, C.: Invariant para-closed subspaces. Indiana Univ. Math. J., 21, 881–907 (1972)
Nordgren, E., Radjabalipour, M., Radjavi, H., Rosenthal, P.: On invariant operator ranges. Trans. Amer. Math. Soc., 251, 389–398 (1979)
Ong, S. C.: Converse of a theorem of Foiaş and reflexive lattices of operator ranges. Indiana Univ. Math. J., 30, 57–63 (1981)
Fillmore, P. A., Longstaff, W. E., MacDonald, G., Radjavi, H., Zhong, Y.: Intersections of nest algebras in finite-dimensions. Linear Alg. & Applic., 350, 185–197 (2002)
Davidson, K. R.: Nest Algebras, Pitman Research Notes in Math., 191, Longman Scientific and Tech. Pub. Co., New York (1988)
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Longstaff, W.E. Small Transitive Families of Subspaces. Acta Math Sinica 19, 567–576 (2003). https://doi.org/10.1007/s10114-003-0267-1
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DOI: https://doi.org/10.1007/s10114-003-0267-1