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