Parallel Addition and Parallel Subtraction of Operators
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 finitedimensional space is considered in the last section.Bibliography: 11 titles.
 Publication:

Izvestiya: Mathematics
 Pub Date:
 April 1976
 DOI:
 10.1070/IM1976v010n02ABEH001694
 Bibcode:
 1976IzMat..10..351P