Relations between (S^ δ/2 T ^γ S^δ/2 ) ^qδ/γ +δ ≥ S^ δq and T^qγ ≥ (T^γ/2 S^δ T^γ/2 )^qγ /γ +δ and their applications

Abstract

Let B^+(H ) represent the cone comprising all positive invertible operators on a complex separable Hilbert space H . When
T and S belong to B^+(H), it holds true that for any γ ≥ 0, δ ≥ 0, and 0 < q ≤ 1, the following two inequalities are equivalent:

                                       (S^ δ/2 T ^γ S^δ/2 ) ^qδ/γ +δ ≥ S^ δq and T^qγ ≥ (T^γ/2 S^δ T^γ/2 )^qγ /γ +δ

In this article, we will explore the connections between these inequalities and provide some applications of this discovery to operator
class theory. Furthermore, we will provide a positive response to the question posed in [16].