A note on the parallel sum

By using a variational principle we find a necessary and sufficient condition for an operator to majorise the parallel sum of two positive definite operators. This result is then used as a vehicle to create new operator inequalities involving the parallel sum.

[1]  R. Subramanian,et al.  Inequalities between means of positive operators , 1978, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  On parallel sum of operators , 2020 .

[3]  R. Duffin,et al.  Series and parallel addition of matrices , 1969 .