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The purpose of this paper is two-fold: we present some matrix inequalities of log-majorization type for eigenvalues indexed by a sequence; we then apply our main theorem to generalize and improve the Hua-Marcus inequalities. Our results are stronger and more general than the existing ones.
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Motivated by Horns log-majorization (singular value) inequality $s(AB)underset{log}{prec} s(A)*s(B)$ and the related weak-majorization inequality $s(AB)underset{w}{prec} s(A)*s(B)$ for square complex matrices, we consider their Hermitian analogs $lambda(sqrt{A}Bsqrt{A}) underset{log}{prec} lambda(A)*lambda(B)$ for positive semidefinite matrices and $lambda(|Acirc B|) underset{w}{prec} lambda(|A|)*lambda(|B|)$ for general (Hermitian) matrices, where $Acirc B$ denotes the Jordan product of $A$ and $B$ and $*$ denotes the componentwise product in $R^n$. In this paper, we extended these inequalities to the setting of Euclidean Jordan algebras in the form $lambdabig (P_{sqrt{a}}(b)big )underset{log}{prec} lambda(a)*lambda(b)$ for $a,bgeq 0$ and $lambdabig (|acirc b|big )underset{w}{prec} lambda(|a|)*lambda(|b|)$ for all $a$ and $b$, where $P_u$ and $lambda(u)$ denote, respectively, the quadratic representation and the eigenvalue vector of an element $u$. We also describe inequalities of the form $lambda(|Abullet b|)underset{w}{prec} lambda({mathrm{diag}}(A))*lambda(|b|)$, 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 H{o}lder type inequality $||acirc b||_pleq ||a||_r,||b||_s$, where $||x||_p:=||lambda(x)||_p$ denotes the spectral $p$-norm of $x$ and $p,q,rin [1,infty]$ with $frac{1}{p}=frac{1}{r}+frac{1}{s}$. We also give precise values of the norms of the Lyapunov transformation $L_a$ and $P_a$ relative to two spectral $p$-norms.
Denote by $P_n$ the set of $ntimes n$ positive definite matrices. Let $D = D_1oplus dots oplus D_k$, where $D_1in P_{n_1}, dots, D_k in P_{n_k}$ with $n_1+cdots + n_k=n$. Partition $Cin P_n$ according to $(n_1, dots, n_k)$ so that $Diag C = C_1oplus dots oplus C_k$. We prove the following weak log majorization result: begin{equation*} lambda (C^{-1}_1D_1oplus cdots oplus C^{-1}_kD_k)prec_{w ,log} lambda(C^{-1}D), end{equation*} where $lambda(A)$ denotes the vector of eigenvalues of $Ain Cnn$. The inequality does not hold if one replaces the vectors of eigenvalues by the vectors of singular values, i.e., begin{equation*} s(C^{-1}_1D_1oplus cdots oplus C^{-1}_kD_k)prec_{w ,log} s(C^{-1}D) end{equation*} is not true. As an application, we provide a generalization of a determinantal inequality of Matic cite[Theorem 1.1]{M}. In addition, we obtain a weak majorization result which is complementary to a determinantal inequality of Choi cite[Theorem 2]{C} and give a weak log majorization open question.
Following the recent work of Jiang and Lin (Linear Algebra Appl. 585 (2020) 45--49), we present more results (bounds) on Harnack type inequalities for matrices in terms of majorization (i.e., in partial products) of eigenvalues and singular values. We discuss and compare the bounds derived through different ways. Jiang and Lins results imply Tungs version of Harnacks inequality (Proc. Amer. Math. Soc. 15 (1964) 375--381); our results %with simpler proofs are stronger and more general than Jiang and Lins. We also show some majorization inequalities concerning Cayley transforms. Some open problems on spectral norm and eigenvalues are proposed.
For $alpha,z>0$ with $alpha e1$, motivated by comparison between different kinds of Renyi divergences in quantum information, we consider log-majorization between the matrix functions begin{align*} P_alpha(A,B)&:=B^{1/2}(B^{-1/2}AB^{-1/2})^alpha B^{1/2}, Q_{alpha,z}(A,B)&:=(B^{1-alphaover2z}A^{alphaover z}B^{1-alphaover2z})^z end{align*} of two positive (semi)definite matrices $A,B$. We precisely determine the parameter $alpha,z$ for which $P_alpha(A,B)prec_{log}Q_{alpha,z}(A,B)$ and $Q_{alpha,z}(A,B)prec_{log}P_alpha(A,B)$ holds, respectively.
We survey some classical norm inequalities of Hardy, Kallman, Kato, Kolmogorov, Landau, Littlewood, and Rota of the type [ |A f|_{mathcal{X}}^2 leq C |f|_{mathcal{X}} big|A^2 fbig|_{mathcal{X}}, quad f in dombig(A^2big), ] and recall that under exceedingly stronger hypotheses on the operator $A$ and/or the Banach space $mathcal{X}$, the optimal constant $C$ in these inequalities diminishes from $4$ (e.g., when $A$ is the generator of a $C_0$ contraction semigroup on a Banach space $mathcal{X}$) all the way down to $1$ (e.g., when $A$ is a symmetric operator on a Hilbert space $mathcal{H}$). We also survey some results in connection with an extension of the Hardy-Littlewood inequality involving quadratic forms as initiated by Everitt.
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