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Parallel Sum of Bounded Operators with Closed Ranges

Author

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  • Wenting Liang

    (School of Mathematics and Statistics, Taiyuan Normal University, Jinzhong 030619, China)

Abstract

Let H be a separable infinite dimensional complex Hilbert space and B ( H ) be the set of all bounded linear operators on H . In this paper, we present several conditions under which the distributive law of the parallel sum is valid. It is proved that the parallel sum for positive operators with closed ranges is continued at 0. For A , B ∈ B ( H ) with closed ranges, it is proved that A ≤ ¯ B if and only if A and B − A are parallel summable with the parallel sum A : ( B − A ) = 0 , where the symbol “ ≤ ¯ ” denotes the minus partial order.

Suggested Citation

  • Wenting Liang, 2023. "Parallel Sum of Bounded Operators with Closed Ranges," Mathematics, MDPI, vol. 11(13), pages 1-18, June.
    • Handle: RePEc:gam:jmathe:v:11:y:2023:i:13:p:2897-:d:1181656
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