Fumio Hiai, Takashi Sano
The matrix convexity and the matrix monotony of a real $C^1$ function $f$ on $(0,\infty)$ are characterized in terms of the conditional negative or positive definiteness of the Loewner matrices associated with $f$, $tf(t)$, and $t^2f(t)$. Similar characterizations are also obtained for matrix monotone functions on a finite interval $(a,b)$.
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We study upper bounds on the Schur multiplier norm of Loewner matrices for concave and convex functions. These bounds then immediately lead to upper bounds on the ratio of Schatten $q$-norms of commutators $||\;[A,f(B)]\;||_q/||\;[A,B]\;||_q$. We also consider operator monotone functions, for which sharper bounds are obtained.
The concept of Loewner (partial) order for general complex matrices is introduced. After giving the definition of arithmetic, geometric, and harmonic mean for accretive-dissipative matrices, we study their basic properties. An AM-GM-HM inequality is established for two accretive-dissipative matrices in the sense of this extended Loewner order. We also compare the harmonic mean and parallel sum of two accretive-dissipative matrices, revealing an interesting relation between them. A number of examples are included.
The classical Loewner's theorem states that operator monotone functions on real intervals are described by holomorphic functions on the upper half-plane. We characterize local order isomorphisms on operator domains by biholomorphic automorphisms of the generalized upper half-plane, which is the collection of all operators with positive invertible imaginary part. We describe such maps in an explicit manner, and examine properties of maximal local order isomorphisms. Moreover, in the finite-dimensional case, we prove that every order embedding of a matrix domain is a homeomorphic order isomorphi…
Entrywise powers of symmetric matrices preserving positivity, monotonicity or convexity with respect to the Loewner ordering arise in various applications, and have received much attention recently in the literature. Following FitzGerald and Horn [J. Math. Anal. Appl., 1977], it is well-known that there exists a critical exponent beyond which all entrywise powers preserve positive definiteness. Similar phase transition phenomena have also recently been shown by Hiai (2009) to occur for monotonicity and convexity. In this paper, we complete the characterization of all the entrywise powers below…
Entrywise Loewner Preservers on Min and Max Matrix Cones
本文刻画了Min与Max矩阵锥上按元素Loewner保持映射的性质，明确了函数保持半正定性、全非负性等的条件，还建立了严格锥与相关矩阵类的关联。
We extend the study of the Pick class, the set of complex analytic functions taking the upper half plane into itself, to the noncommutative setting. R. Nevanlinna showed that elements of the Pick class have certain integral representations which reflect their asymptotic behavior at infinity. Loewner connected the Pick class to matrix monotone functions. We generalize the Nevanlinna representation theorems and Loewner's theorem on matrix monotone functions to the free Pick class, the collection of functions that map tuples of matrices with positive imaginary part into the matrices with positive…
We give new characterizations for matrix monotonicity and convexity of fixed order which connects previous characterizations by Loewner, Dobsch, Donoghue, Kraus and Bendat--Sherman. The ideas introduced are then used to characterize matrix monotone functions of arbitrary order on general subsets of the real line.
Entrywise functions preserving Loewner positivity have been studied by many authors, most notably Schoenberg and Rudin. Following their work, it is known that functions preserving positivity when applied entrywise to positive semidefinite matrices of all dimensions are necessarily analytic with nonnegative Taylor coefficients. When the dimension is fixed, it has been shown by Vasudeva and Horn that such functions are automatically continuous and sufficiently differentiable. A natural refinement of the aforementioned problem consists of characterizing functions preserving positivity under rank …
We introduce and investigate the resolvent order, which is a binary relation on the set of firmly nonexpansive mappings. It unifies well-known orders introduced by Loewner (for positive semidefinite matrices) and by Zarantonello (for projectors onto convex cones). A connection with Moreau's order of convex functions is also presented. We also construct partial orders on (quotient sets of) proximal mappings and convex functions. Various examples illustrate our results.
An algorithm to exactly compute minimal upper bounds in the Loewner order
提出一种迭代算法，能在最多n次迭代内精确计算任意有限个n×n Hermitian矩阵在Loewner序下的最小上界，并给出Python实现。
Bi-monotone maps on the set of all variance-covariance matrices with respect to minus partial order
本文刻画了在半正定实矩阵锥上双向保持减偏序的所有满射形式，完整解决了该偏序保持映射的分类问题。
Towards a Characterization of Matrix Monotonicity via Radial Convexity
该研究针对正定矩阵锥上的矩阵序引入度量的径向凸性，关联其与Loewner矩阵Hadamard平方正定性的关系，在2次情形证明n阶径向凸性等价于n次矩阵单调性，还得到3次情形刻画并讨论推广。
We generalize Loewner's method for proving that matrix monotone functions are operator monotone. The relation x \leq y on bounded operators is our model for a definition for C*-relations of being residually finite dimensional. Our main result is a meta-theorem about theorems involving relations on bounded operators. If we can show there are residually finite dimensional relations involved, and verify a technical condition, then such a theorem will follow from its restriction to matrices. Applications are shown regarding norms of exponentials, the norms of commutators and "positive" noncommutat…
We establish local characterizations of matrix monotonicity and convexity of fixed order by giving integral representations connecting the Loewner and Kraus matrices, previously known to characterize these properties, to respective Hankel matrices. Our results are new already in the general case of matrix convexity and our approach significantly simplifies the corresponding work on matrix monotonicity. We also obtain an extension of the original characterization for matrix convexity by Kraus, and tighten the relationship between monotonicity and convexity.
Fe{j}ér and Levin-Stečkin inequalities treat integrals of the product of convex functions with symmetric functions. The main goal of this article is to present possible matrix versions of these inequalities. In particular, majorization results are shown of Fejér type for both convex and log-convex functions. For matrix Levin-Stečkin type, we present more rigorous results involving the partial Loewner ordering for Hermitian matrices.
Kubo Ando Means on the Cone of J Hermitian Matrices
本文将Kubo-Ando均值理论扩展到正四元数厄米特矩阵锥，并针对J-厄米特正矩阵锥建立类似框架，证明其均值与算子单调函数一一对应，扩展了经典特征到不定情形。
Functions preserving Loewner positivity when applied entrywise to positive semidefinite matrices have been widely studied in the literature. Following the work of Schoenberg [Duke Math. J. 9], Rudin [Duke Math. J. 26], and others, it is well-known that functions preserving positivity for matrices of all dimensions are absolutely monotonic (i.e., analytic with nonnegative Taylor coefficients). In this paper, we study functions preserving positivity when applied entrywise to sparse matrices, with zeros encoded by a graph $G$ or a family of graphs $G_n$. Our results generalize Schoenberg and Rudi…
We characterize real functions $f$ on an interval $(-α,α)$ for which the entrywise matrix function $[a_{ij}] \mapsto [f(a_{ij})]$ is positive, monotone and convex, respectively, in the positive semidefiniteness order. Fractional power functions are exemplified and related weak majorizations are shown.
$q$-Numerical Radius Estimates in Semi-Hilbertian Spaces and Their Relations with Matrix Means for Sectorial Matrices
本文研究了半希尔伯特空间中算子的$q$-数值半径，推导出新的特征化结果及精确上界和下界，并获得涉及算子单调函数和扇形矩阵矩阵均值的不等式。
For positive definite matrices $A$ and $B$, the Kubo-Ando matrix power mean is defined as $$ P_μ(p, A, B) = A^{1/2}\left(\frac{1+(A^{-1/2}BA^{-1/2})^p}{2}\right )^{1/p} A^{1/2}\quad (p \ge 0). $$ In this paper, for $0\le p \le 1 \le q$, we show that if one of the following inequalities \begin{align*} f(P_μ(p, A, B)) \le f(P_μ(1, A, B)) \le f(P_μ(q, A, B))\nonumber \end{align*} holds for any positive definite matrices $A$ and $B$, then the function $f$ is operator monotone on $(0, \infty).$ We also study the inverse problem for non-Kubo-Ando matrix power means with the powers $1/2$ and $2$. As …
We prove generalizations of Löwner's results on matrix monotone functions to several variables. We give a characterization of when a function of $d$ variables is locally monotone on $d$-tuples of commuting self-adjoint $n$-by-$n$ matrices. We prove a generalization to several variables of Nevanlinna's theorem describing analytic functions that map the upper half-plane to itself and satisfy a growth condition. We use this to characterize all rational functions of two variables that are operator monotone.
Preserving positivity for rank-constrained matrices
本文通过引入秩约束，在固定维度下刻画了将秩至多为l的半正定矩阵映射为秩至多为k的半正定矩阵的逐元素函数，推广了Horn和Loewner的必要条件，并给出了经典全维度正性保持定理的初等证明。
Functional Calculi, Positivity, and Convolution of Matrices
本文提出了一种基于矩阵卷积的函数 calculus 框架，揭示了卷积与对称群 Bruhat 秩的联系，并扩展了经典正性保持理论。
For positive semidefinite matrices $A$ and $B$, Ando and Zhan proved the inequalities $||| f(A)+f(B) ||| \ge ||| f(A+B) |||$ and $||| g(A)+g(B) ||| \le ||| g(A+B) |||$, for any unitarily invariant norm, and for any non-negative operator monotone $f$ on $[0,\infty)$ with inverse function $g$. These inequalities have very recently been generalised to non-negative concave functions $f$ and non-negative convex functions $g$, by Bourin and Uchiyama, and Kosem, respectively. In this paper we consider the related question whether the inequalities $||| f(A)-f(B) ||| \le ||| f(|A-B|) |||$, and $||| g(A…
Given a function $f: (a,b) \rightarrow \mathbb{R},$ Löwner's theorem states $f$ is monotone when extended to self-adjoint matrices via the functional calculus, if and only if $f$ extends to a self-map of the complex upper half plane. In recent years, several generalizations of Löwner's theorem have been proven in several variables. We use the relaxed Agler, McCarthy and Young theorem on locally matrix monotone functions in several commuting variables to generalize results in the noncommutative case. Specifically, we show that a real free function defined over an operator system must analytical…
Accretive Partial Transpose Matrices and Their Connections to Matrix Means
本文研究增生部分转置（APT）矩阵的性质，将其推广自PPT矩阵，并证明在扇形矩阵和算子单调函数条件下，由加权算术均值构造的块矩阵保持APT性质。
Some real functions f induce mean of positive numbers and the matrix monotonicity gives a possibility for means of positive definite matrices. Moreover, such a function f can define linear mapping beta on matrices (which is basic in the constructions of monotone metrics). The present subject is to check the complete positivity of beta in the case of a few concrete functions f. This problem has been motivated by applications in quantum information.
Positive Semidefinite Matrix Supermartingales
本文研究半正定矩阵超鞅与后向子鞅的收敛性及极大不等式，并据此给出随机对称矩阵在多种尾部条件下的新型集中不等式。
Let $\mathcal{C}_{d,n}$ be the convex cone consisting of real $n$-variate degree $d$ forms that are strictly positive on $\mathbb{R}^n\setminus \{\mathbf{0}\}$. We prove that the Lebesgue volume of the sublevel set $\{g\leq 1\}$ of $g\in \mathcal{C}_{d,n}$ is a completely monotone function on $\mathcal{C}_{d,n}$ and investigate the related properties. Furthermore, we provide (partial) characterization of forms, whose sublevel sets have finite Lebesgue volume. Finally, we discover an interesting property of a centered Gaussian distribution, establishing a connection between the matrix of its de…
Given a convex function $φ$ and two hermitian matrices $A$ and $B$, Lewin and Sabin study in [M. Lewin, J. Sabin, {\it A Family of Monotone Quantum Relative Entropies}, Lett. Math. Phys. \textbf{104} (2014), 691-705.] the relative entropy defined by $\mathcal{H}(A,B)=\text{Tr} [ φ(A) - φ(B) - φ'(B)(A-B) ]$. Amongst other things, they prove that the so-defined quantity is monotone if and only if $φ'$ is operator monotone. The monotonicity is then used to properly define $\mathcal{H}(A,B)$ for self-adjoint bounded operators acting on an infinite-dimensional Hilbert space by a limiting procedure.…
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