2020/03/27 by Tao, Jiyuan, Jeong, Juyoung, Gowda, Muddappa
#15A33 #17C20 #17C55 #FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2003.12377
Motivated by Horn's log-majorization (singular value) inequality s(AB)\undersetlog\prec s(A)*s(B) and the related weak-majorization inequality s(AB)\undersetw\prec s(A)*s(B) for square complex matrices, we consider their Hermitian analogs λ(√(A)B√(A)) \undersetlog\prec λ(A)*λ(B) for positive semidefinite matrices and λ(|A∘ B|) \undersetw\prec λ(|A|)*λ(|B|) for general (Hermitian) matrices, where A∘ B denotes the Jordan product of A and B and * denotes the componentwise product in Rn. In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form λ (P√(a)(b) )\undersetlog\prec λ(a)*λ(b) for a,b≥ 0 and λ (|a∘ b| )\undersetw\prec λ(|a|)*λ(|b|) for all a and b, where Pu and λ(u) denote, respectively, the quadratic representation and the eigenvalue vector of an element u. We also describe inequalities of the form λ(|A\bullet b|)\undersetw\prec λ(diag(A))*λ(|b|), where A is a real symmetric positive semidefinite matrix and A \bullet b is the Schur product of A and b. In the form of an application, we prove the generalized Hölder type inequality ||a∘ b||p≤ ||a||r ||b||s, where ||x||p:=||λ(x)||p denotes the spectral p-norm of x and p,q,r∈ [1,∞] with (1)/(p)=(1)/(r)+(1)/(s). We also give precise values of the norms of the Lyapunov transformation La and Pa relative to two spectral p-norms.