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This article is cited in 23 scientific papers (total in 23 papers)
Parallel addition and parallel subtraction of operators
È. L. Pekarev, Yu. L. Shmul'yan
Abstract:
The parallel sum $A:B$ of two invertible nonnegative operators $A$ and $B$ in a Hilbert space $\mathfrak H$ is the operator $(A^{-1}+B^{-1})^{-1}=A(A+B)^{-1}B$. This definition was extended to noninvertible operators by Anderson and Duffin for the case $\dim\mathfrak H<\infty$ and by Fillmore and Williams for the general case.
The investigation of parallel addition is continued in this paper; in particular, associativity is proved.
Criteria are established for solvability of the equation $A:X=S$ with an unknown operator $X$ when $A$ and $S$ are given. In the case of solvability, the existence of a minimal solution $S\div A$, called the parallel difference, is proved.
Parallel subtraction in a finite-dimensional space is considered in the last section.
Bibliography: 11 titles.
Received: 11.04.1974
Citation:
È. L. Pekarev, Yu. L. Shmul'yan, “Parallel addition and parallel subtraction of operators”, Izv. Akad. Nauk SSSR Ser. Mat., 40:2 (1976), 366–387; Math. USSR-Izv., 10:2 (1976), 351–370
Linking options:
https://www.mathnet.ru/eng/im2115https://doi.org/10.1070/IM1976v010n02ABEH001694 https://www.mathnet.ru/eng/im/v40/i2/p366
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Abstract page: | 734 | Russian version PDF: | 182 | English version PDF: | 14 | References: | 63 | First page: | 1 |
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