Research Article

Parallel sum

Published in: Quaestiones Mathematicae
Volume 48 , issue 3 : Entrepreneurship and Africa’s Cultural Context , 2024 , pages: 475–484
DOI: 10.2989/16073606.2024.2400958
Author(s): Stefan Stankov University of Niš, Serbia,
Keywords: Primary: 15A09, Secondary: 47A05, Parallel sum, operators,
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Abstract

In this paper, we will consider parallel summable operators on Hilbert space. Operators A and B are said to be parallel summable if and . We will prove that the parallel sum can be represented as A: B = A(A + B)† B without additional assumption of the closedness of the range of the operator A + B. Furthermore, we will derive the equalities C (A : B) = C A : C B and (A : B) : C = A : (B : C ) under weaker conditions than the ones represented in [X. Tian, S. Wang, C. Deng, On parallel sum of operators, Linear Algebra Appl. 603 (2020) 57–83] and [W. Luo, C. Song, Q. Xu, The parallel sum for adjointable operators on Hilbert C* -modules, Acta Math. Sin. 62 (2019) 541–552]. Finally, we will extend some recent result for Hermitian positive semi-definite matrices to bounded linear operators.
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