The parallel sum of nonlinear monotone operators. (English) Zbl 0628.47033

Let H be a Hilbert space with inner product \(<*,.>\) and operators on H are identified with their graphs. A monotone operator A on H is a subset of \(H\times H\) such that for any \([x_ i,y_ i]\in A\), \(i=1,2\), we have \(<y_ 1-y_ 2,x_ 1-x_ 2>\geq 0\). The parallel sum of A and B denoted by A: B is defined by \((A^{-1}+B^{-1})^{-1}.\)
The author discusses algebraic properties of the parallel sums and shows that the class of monotone operators is closed under the operation of parallel sums. The question of determination of the domain of the parallel sum is looked into. A few results are sharpened by considering projections and subdifferentials.
Reviewer: N.K.Thakare


47H05 Monotone operators and generalizations
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[1] Anderson, W. N., Shorted operators, SIAM J. appl. Math., 20, 520-525 (1971) · Zbl 0217.05503
[2] Anderson, W. N.; Duffin, R. J., Series and parallel addition of matrices, J. math. Analysis Applic., 26, 576-594 (1969) · Zbl 0177.04904
[3] Anderson, W. N.; Morley, T. D.; Trapp, G. E., Fenchel duality of nonlinear networks, IEEE Trans. Circuits Systems, 9, 762-765 (1978) · Zbl 0392.94016
[4] Anderson, W. N.; Schreiber, M., The infimum of two projections, Acta Sci. Math. (Szeged), 33, 165-168 (1972) · Zbl 0258.46023
[5] Anderson, W. N.; Trapp, G. E., Shorted operators II, SIAM J. appl. Math., 28, 60-71 (1975) · Zbl 0295.47032
[6] Brezis, H., New results concerning monotone operators and nonlinear semigroups, Proc. Symp. on Functional and Numerical Analysis of Nonlinear Problems (1974), Kyoto
[7] Brezis, H., Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert (1973), North-Holland: North-Holland Amsterdam · Zbl 0252.47055
[8] Brezis, H.; Haraux, A., Image d’une somme d’operateurs monotones et applications, Israel J. Math., 23, 165-186 (1976) · Zbl 0323.47041
[9] Dolezal, V., Monotone Operators and Applications in Control and Network Theory (1979), Elsevier Scientific: Elsevier Scientific Amsterdam · Zbl 0425.93002
[10] Fillmore, P. A.; Williams, J. P., On operator ranges, Adv. Math., 7, 254-281 (1971) · Zbl 0224.47009
[11] Kinderlehrer, D.; Stampacchia, G., An Introduction to Variational Inequalities and Their Applications (1980), Academic Press: Academic Press New York · Zbl 0457.35001
[12] Kubo, F., Conditional expectations and operations derived from network connections, J. math. Analysis Applic., 80, 477-489 (1981) · Zbl 0489.94023
[13] Morley, T. D., Parallel summation, Maxwell’s principle and the infimum of projections, J. Math. Analysis Applic., 70, 33-41 (1979) · Zbl 0456.47021
[14] Rockafellar, R. T., Characterization of the subdifferentials of convex functions, Pacif. J. Math., 17, 497-510 (1966) · Zbl 0145.15901
[15] Rockafellar, R. T., Convex Analysis (1979), Princeton University Press: Princeton University Press Princeton, New Jersey · Zbl 0419.90024
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