Representing a closed operator as a quotient of continuous operators

Kaufman, William E.

doi:10.1090/S0002-9939-1978-0509249-9

Representing a closed operator as a quotient of continuous operators
HTML articles powered by AMS MathViewer

by William E. Kaufman

Proc. Amer. Math. Soc. 72 (1978), 531-534

DOI: https://doi.org/10.1090/S0002-9939-1978-0509249-9

PDF | Request permission

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

P. A. Fillmore and J. P. Williams, On operator ranges, Advances in Math. 7 (1971), 254–281. MR 293441, DOI 10.1016/S0001-8708(71)80006-3
J. J. Koliha, Convergent and stable operators and their generalization, J. Math. Anal. Appl. 43 (1973), 778–794. MR 324447, DOI 10.1016/0022-247X(73)90292-8
J. S. MacNerney, Investigation concerning positive definite continued fractions, Duke Math. J. 26 (1959), 663–677. MR 117326
J. S. MacNerney, Continuous embeddings of Hilbert spaces, Rend. Circ. Mat. Palermo (2) 19 (1970), 109–112. MR 303267, DOI 10.1007/BF02843890
Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727
J. von Neumann, Über adjungierte Funktionaloperatoren, Ann. of Math. (2) 33 (1932), no. 2, 294–310 (German). MR 1503053, DOI 10.2307/1968331

Functional operators