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The factorization of a linear conjugate symmetric involution in Hilbert space

Author: James W. Moeller
Journal: Proc. Amer. Math. Soc. 94 (1985), 269-272
MSC: Primary 47B25
Erratum: Proc. Amer. Math. Soc. 97 (1986), 568.
MathSciNet review: 784177
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Abstract: Let $ X$ be a closed linear transformation whose domain is dense in the complex separable Hilbert space $ H$ and whose adjoint is denoted by $ {X^ * }$. The operator $ X$ is said to be conjugate symmetric if $ \Gamma (X) \subset \Gamma (Q{X^ * }Q)$, where $ \Gamma (X)$ represents the graph of $ X$ in $ H \otimes H$ and $ Q$ is a conjugation on $ H$. The main theorem in this note states that a conjugate symmetric linear involution $ X$ satisfies the operator equation $ X = Q{X^ * }Q$.

References [Enhancements On Off] (What's this?)

  • [1] N. Dunford and J. T. Schwartz, Linear operators. Part II: Spectral theory, Wiley, New York, 1963. MR 1009163 (90g:47001b)
  • [2] L. M. Falicov, Group theory and its physical applications, Univ. of Chicago Press, Chicago, Ill., 1966. MR 0205615 (34:5441)
  • [3] T. Kato, Perturbation theory for linear operators, Springer-Verlag, New York, 1966. MR 0203473 (34:3324)
  • [4] J. Moeller, A double commutant theorem for conjugate selfadjoint operators, Proc. Amer. Math. Soc. 83 (1981), 506-508. MR 627679 (84d:47044)
  • [5] J. von Neumann, Uber adjungierte Funktionaloperatoren, Ann. of Math. 33 (1932), 294-310. MR 1503053
  • [6] H. Radjavi and P. Rosenthal, Invariant subspaces, Springer-Verlag, New York, 1973. MR 0367682 (51:3924)
  • [7] F. Riesz and B. Sz.-Nagy, Functional analysis, Ungar, New York, 1955. MR 0071727 (17:175i)

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Keywords: Hilbert space, closed linear transformation, conjugation operator
Article copyright: © Copyright 1985 American Mathematical Society

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