Abstract
We study two subspace systems in a separable infinite-dimensional Hilbert space up to (bounded) isomorphism. One of the main result of this paper is the following: Isomorphism classes of two subspace systems given by graphs of bounded operators are determined by unitarily equivalent classes of the operator ranges and the nullity of the original bounded operators giving graphs. We construct several non-isomorphic examples of two subspace systems in an infinite-dimensional Hilbert space. Even if we study n subspace systems for \(n \ge 3\), we can use the analysis of any two subspaces of the n subspaces.
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References
Araki, H.: A lattice of von Neumann algebras associated with the quantum theory of a free Bose field. J. Math. Phys. 4, 1343 (1963)
Davis, C.: Separation of two linear subspaces. Acta Sci. Math. (Szeged) 1(9), 172–187 (1958)
Dixmier, J.: Position relative de deux variétés linéaires fermées dans un espace de Hilbert. Rev. Sci. 86, 387–399 (1948)
Enomoto, M., Watatani, Y.: Relative position of four subspaces in a Hilbert space. Adv. Math. 201, 263–317 (2006)
Enomoto, M., Watatani, Y.: Indecomposable representations of quivers on infinite-dimensional Hilbert spaces. J. Funct. Anal. 256, 959–991 (2009)
Enomoto, M., Watatani, Y.: Relative position of three subspaces in a Hilbert space. Acta Sci. Math. (Szeged) 85, 519–537 (2019)
Fillmore, P., Williams, J.: On operator ranges. Adv. Math. 7, 254–281 (1971)
Halmos, P.R.: Two subspaces. Trans. Am. Math. Soc. 144, 381–389 (1969)
Kruglyak, S., Rabanovich, V., Samoilenko, Y.: On sums of projections. Funct. Anal. Appl. 36, 182–195 (2002)
Kruglyak, S., Samoilenko, Y.: On the complexity of description of representations of \(*\)-algebras generated by idempotents. Proc. Am. Math. Soc. 128, 1655–1664 (2000)
Lacey, H.E.: The Hamel dimension of any infinite dimensional separable Banach space is c. Am. Math. Mon. 80, 298 (1973)
McCarthy, C.A.: \(c_{p}\). Isr. J. Math. 5, 249–271 (1967)
Moskaleva, Y., Samoilenko, Y.: Systems of \(n\) subspaces and representations of *-algebras generated by projections. Methods Funct. Anal. Topol. 12, 57–73 (2006)
Stone, M.H.: On unbounded operators in Hilbert space. J. Indian Math. Soc. 15, 155–192 (1951)
Sunder, V.S.: N-subspaces. Can. J. Math. 40, 38–54 (1988)
Acknowledgements
This work was supported by JSPS KAKENHI Grant Numbers JP 23654053 and JP 17K18739. This work was also supported by the Research Institute for Mathematical Sciences, a Joint Usage/Research Center located in Kyoto University.
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Communicated by Hiroyuki Osaka.
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Enomoto, M., Watatani, Y. Systems of two subspaces in a Hilbert space. Adv. Oper. Theory 6, 4 (2021). https://doi.org/10.1007/s43036-020-00109-y
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DOI: https://doi.org/10.1007/s43036-020-00109-y