Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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
DOI: https://doi.org/10.1090/S0002-9939-1978-0509249-9
MathSciNet review: 509249
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


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

  • [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.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47A65

Retrieve articles in all journals with MSC: 47A65


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0509249-9
Keywords: Closed operator, Hilbert space, complete inner product space, strictly contractive operator, quotients of operators, Cayley transform
Article copyright: © Copyright 1978 American Mathematical Society

American Mathematical Society