RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Izv. RAN. Ser. Mat.: Year: Volume: Issue: Page: Find

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

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)
References: PDF file   HTML file

English version:
Mathematics of the USSR-Izvestiya, 1976, 10:2, 351–370

Bibliographic databases:

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

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} `

• http://mi.mathnet.ru/eng/izv2115
• http://mi.mathnet.ru/eng/izv/v40/i2/p366

 SHARE:

Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
2. George E. Trapp, “Hermitian semidefinite matrix means and related matrix inequalities—an introduction”, Linear and Multilinear Algebra, 16:1-4 (1984), 113
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
4. Marie-Laurence mazure, “La soustraction parallèle d’operateurs Interprétée comme déconvolution de formes quadratiques convexes”, Optimization, 18:4 (1987), 465
5. Sirkka-Liisa Eriksson-Bique, Heinz Leutwiler, “A generalization of parallel addition”, Aequ math, 38:1 (1989), 99
6. Alberto Seeger, “Direct and inverse addition in convex analysis and applications”, Journal of Mathematical Analysis and Applications, 148:2 (1990), 317
7. Jonathan Arazy, “Operator means and networks”, Integr equ oper theory, 14:2 (1991), 157
8. M. -L. Mazure, “Equations de convolution et formes quadratiques”, Annali di Matematica, 158:1 (1991), 75
9. M. Volle, “A formula on the subdifferential of the deconvolution of convex functions”, BAZ, 47:2 (1993), 333
10. J. -B. Hiriart-Urruty, “The deconvolution operation in convex analysis: An introduction”, Cybern Syst Anal, 30:4 (1994), 555
11. G. Corach, A. Maestripieri, D. Stojanoff, “Generalized Schur complements and oblique projections”, Linear Algebra and its Applications, 341:1-3 (2002), 259
12. Jorge Antezana, Gustavo Corach, Demetrio Stojanoff, “Spectral shorted matrices”, Linear Algebra and its Applications, 381 (2004), 197
13. Jorge Antezana, Gustavo Corach, Demetrio Stojanoff, “Bilateral shorted operators and parallel sums”, Linear Algebra and its Applications, 414:2-3 (2006), 570
14. M. Laura Arias, Gustavo Corach, M. Celeste Gonzalez, “Products of projections and positive operators”, Linear Algebra and its Applications, 2013
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
•  Number of views: This page: 559 Full text: 117 References: 50 First page: 1