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Izv. Akad. Nauk SSSR Ser. Mat., 1976, Volume 40, Issue 2, Pages 366–387 (Mi izv2115)  

This article is cited in 15 scientific papers (total in 15 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.

Full text: PDF file (1968 kB)
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English version:
Mathematics of the USSR-Izvestiya, 1976, 10:2, 351–370

Bibliographic databases:

UDC: 513.88
MSC: Primary 47D99; Secondary 47A50, 94A20
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

Citation in format AMSBIB
\Bibitem{PekShm76}
\by \`E.~L.~Pekarev, Yu.~L.~Shmul'yan
\paper Parallel addition and parallel subtraction of operators
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1976
\vol 40
\issue 2
\pages 366--387
\mathnet{http://mi.mathnet.ru/izv2115}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=410429}
\zmath{https://zbmath.org/?q=an:0351.47031}
\transl
\jour Math. USSR-Izv.
\yr 1976
\vol 10
\issue 2
\pages 351--370
\crossref{https://doi.org/10.1070/IM1976v010n02ABEH001694}


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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. È. L. Pekarev, “Convolutions onto an operator domain”, Funct. Anal. Appl., 12:3 (1978), 230–231  mathnet  crossref  mathscinet  zmath
    2. George E. Trapp, “Hermitian semidefinite matrix means and related matrix inequalities—an introduction”, Linear and Multilinear Algebra, 16:1-4 (1984), 113  crossref
    3. William N. Anderson, Jr., Michael E. Mays, Thomas D. Morley, George E. Trapp, “The Contraharmonic Mean of HSD Matrices”, SIAM J Algebraic Discrete Methods, 8:4 (1987), 674  crossref  mathscinet  zmath
    4. Marie-Laurence mazure, “La soustraction parallèle d’operateurs Interprétée comme déconvolution de formes quadratiques convexes”, Optimization, 18:4 (1987), 465  crossref
    5. Sirkka-Liisa Eriksson-Bique, Heinz Leutwiler, “A generalization of parallel addition”, Aequ math, 38:1 (1989), 99  crossref  mathscinet  zmath
    6. Alberto Seeger, “Direct and inverse addition in convex analysis and applications”, Journal of Mathematical Analysis and Applications, 148:2 (1990), 317  crossref
    7. Jonathan Arazy, “Operator means and networks”, Integr equ oper theory, 14:2 (1991), 157  crossref  mathscinet  zmath  isi
    8. M. -L. Mazure, “Equations de convolution et formes quadratiques”, Annali di Matematica, 158:1 (1991), 75  crossref  mathscinet  zmath  isi
    9. M. Volle, “A formula on the subdifferential of the deconvolution of convex functions”, BAZ, 47:2 (1993), 333  crossref
    10. J. -B. Hiriart-Urruty, “The deconvolution operation in convex analysis: An introduction”, Cybern Syst Anal, 30:4 (1994), 555  crossref  mathscinet  zmath
    11. G. Corach, A. Maestripieri, D. Stojanoff, “Generalized Schur complements and oblique projections”, Linear Algebra and its Applications, 341:1-3 (2002), 259  crossref
    12. Jorge Antezana, Gustavo Corach, Demetrio Stojanoff, “Spectral shorted matrices”, Linear Algebra and its Applications, 381 (2004), 197  crossref
    13. Jorge Antezana, Gustavo Corach, Demetrio Stojanoff, “Bilateral shorted operators and parallel sums”, Linear Algebra and its Applications, 414:2-3 (2006), 570  crossref
    14. M. Laura Arias, Gustavo Corach, M. Celeste Gonzalez, “Products of projections and positive operators”, Linear Algebra and its Applications, 2013  crossref
    15. M. Laura Arias, Gustavo Corach, Alejandra Maestripieri, “Range additivity, shorted operator and the Sherman–Morrison–Woodbury formula”, Linear Algebra and its Applications, 467 (2015), 86  crossref
  • Известия Академии наук СССР. Серия математическая Izvestiya: Mathematics
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