Affiliation:
1. School of Mathematics and Statistics, Taiyuan Normal University, Jinzhong 030619, China
Abstract
Let H be a separable infinite dimensional complex Hilbert space and B(H) be the set of all bounded linear operators on H. In this paper, we present several conditions under which the distributive law of the parallel sum is valid. It is proved that the parallel sum for positive operators with closed ranges is continued at 0. For A,B∈B(H) with closed ranges, it is proved that A≤¯B if and only if A and B−A are parallel summable with the parallel sum A:(B−A)=0, where the symbol “≤¯” denotes the minus partial order.
Funder
National Natural Science Foundation of China
Fundamental Research Program of Shanxi Province
Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi
Subject
General Mathematics,Engineering (miscellaneous),Computer Science (miscellaneous)
Reference19 articles.
1. Network synthesis through hybrid matrices;Duffin;SIAM Appl. Math.,1966
2. A new operation for analyzing series-parallel networks;Erickson;IEEE Trans. Circuit Theory CT-6,1959
3. Shorted operator;Anderson;SIAM J. Appl. Math.,1971
4. On parallel summability of matrices;Mitra;Linear Algebra Appl.,1986
5. Bilateral shorted operators and parallel sums;Antezana;Linear Algebra Appl.,2006