Note on the assumptions in working with generalized inverses

摘要

Most of the published results on solving operator equations are very restrictive i.e. they have been proved under certain additional assumptions and in fact we do not have general solvability conditions. There are many reasons why this is so, one of them being the fact that the usual methods employed when solving these equations involve the use of generalized inverses which exist and are bounded in the case of operators only under certain special conditions such as closedness of the range of operators. Using two previously considered systems of operator equations as examples, we will show that using a particular general approach we can give necessary and sufficient solvability conditions without any additional assumptions. We will consider the system and generalize recent particular results from the paper C. Deng, et al. (Appl. Math. Letters 81, 86–92 (2018)), as well as for which the particular results are given in the paper C. Deng, (J. Math. Anal. Appl. 398 (2013) 664–670). We intend to use our approach to initiate formulating various general solvability conditions for other systems of operator equations.