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Proceedings of the American Mathematical Society

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Representing a closed operator as a quotient of continuous operators

Author: William E. Kaufman
Journal: Proc. Amer. Math. Soc. 72 (1978), 531-534
MSC: Primary 47A65
MathSciNet review: 509249
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Abstract: The closed operators in a Hilbert space H are characterized as quotients $ A{B^{ - 1}}$ of continuous operators on H such that the vector sum $ {A^\ast}(H) + {B^\ast}(H)$ is closed. This leads to the function $ \Gamma (A) = A{(1 - {A^\ast}A)^{ - 1/2}}$, which is shown to map the strictly contractive operators on H reversibly onto the closed densely-defined operators, so as to preserve the selfadjoint and nonnegative conditions.

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  • [1] P. A. Fillmore and J. P. Williams, On operator ranges, Advances in Math. 7 (1971), 254-271. MR 0293441 (45:2518)
  • [2] J. J. Koliha, Convergent and stable operators and their generalizations, J. Math. Anal. Appl. 43 (1973), 778-794. MR 0324447 (48:2799)
  • [3] J. S. Mac Nerney, Investigation concerning positive definite continued fractions, Duke Math. J. 26 (1959), 663-678. MR 0117326 (22:8107)
  • [4] -, Continuous embeddings of Hilbert spaces, Rend. Circ. Mat. Palmero (2) 19 (1970), 109-112. MR 0303267 (46:2405)
  • [5] F. Riesz and B. Sz.-Nagy, Functional analysis, Ungar, New York, 1955 (transl, of 2nd French ed., 1952). MR 0071727 (17:175i)
  • [6] J. von Neumann, Über adjungierte Funktionaloperatoren, Ann. of Math. (2) 33 (1932), 294-310. MR 1503053
  • [7] J. von Neumann, Functional operators, vol. II, Ann. of Math. Studies, no. 22, Princeton Univ. Press, Princeton, N. J., 1950.

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Keywords: Closed operator, Hilbert space, complete inner product space, strictly contractive operator, quotients of operators, Cayley transform
Article copyright: © Copyright 1978 American Mathematical Society

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