Adjointations of Operator Inequalities for Sector Matrices

In this paper, we first extend the well-known inequalities to the case of sector matrices. We also explore the adjointness of operator inequalities with binary operations for sector matrices. As a result of our exploration, we establish four distinct inequalities: a matrix inequality, a unitarily invariant norm inequality, a singular value inequality, and a determinant inequality. For example, we demonstrate that if σ1 and σ2 are non-zero connections, and if A, B, and C belong to Sα, such that R (Aσ1(Bσ2C)) ≤ cos⁴ (α) R ((Aσ1B)σ2(Aσ1C)), then R (Aσ∗1 (Bσ∗2C)) ≥ cos⁴ (α) R ((Aσ∗1B)σ∗2 (Aσ∗1C)).