Summary
Parallel addition of positive operators, a concept introduced by W. N. Anderson and R. J. Duffin [1] in connection with network theory, has already been studied by several authors. We specifically mention W. N. Anderson and G. E. Trapp [2] and [3], T. Ando [4], K. Nishio [12], E. L. Pekarev and J. L. Smul'jan [13], as well as the article [10] by the present authors.
The purpose of this note is to study a generalization of parallel addition. In particular, it will be shown (Theorem 3.2) that the corresponding quasi-units, a concept introduced in [10], are again the extreme points of the convex sets, formed by the positive operators less than or equal to some fixed operator.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Anderson, W. N. andDuffin, R. J.,Series and parallel addition of matrices. J. Math. Anal. Appl.26, 576–594 (1969).
Anderson, W. N. andTrapp, G. E.,Shorted operators, II. SIAM J. Appl. Math.28 (1975), 60–71.
Anderson, W. N. andTrapp, G. E.,The extreme points of a set of positive semidefinite operators. (Preprint).
Ando, T.,Lebesgue-type decomposition of positive operators. Acta Sci. Math.38 (1976), 253–260.
Ando, T. andNishio, K.,Characterizations of operations derived from network connections. J. Math. Analysis and Appl.53 (1976), 539–549.
Arsove, M. G. andLeutwiler, H.,Infinitesimal generators and quasi-units in potential theory. Proc. Nat. Acad. Sci. U.S.A.72 (1975), 2498–2500.
Arsove, M. G. andLeutwiler, H.,Algebraic potential theory. Mem. Amer. Math. Soc.23 (1980), no. 226.
Arsove, M. G. andLeutwiler, H.,A unified theory of harmonic measures and capacitary potentials. Math. Z.183 (1983), 419–442.
Bohnenblust, F.,An axiomatic characterization of L p spaces. Duke Math. J.6 (1940), 627–640.
Eriksson, S. L. andLeutwiler, H.,A potential-theoretic approach to parallel addition. Math. Ann.274 (1986), 301–317.
Luxemburg, W. A. J. andZaanen, A. C.,Riesz spaces, Vol. I. North-Holland, American Elsevier, New York, 1971.
Nishio, K.,Characterization of Lebesque-type decomposition of positive operators. Acta Sci. Math. (Szeged)42 (1980), 145–152.
Pekarev, E. L. andSmul'jan, J. L.,Parallel addition and parallel subtraction of operators. Math. U.S.S.R. Izv.10 (1976), 351–370.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Eriksson-Bique, SL., Leutwiler, H. A generalization of parallel addition. Aeq. Math. 38, 99–110 (1989). https://doi.org/10.1007/BF01839498
Received:
Issue date:
DOI: https://doi.org/10.1007/BF01839498
