Monotonicity for entrywise functions of matrices

We characterize real functions $f$ on an interval $(-\alpha,\alpha)$ 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.Comment: 23 pages; Section 6 is considerably improve

Matrix limit theorems of Kato type related to positive linear maps and operator means

We obtain limit theorems for $\Phi(A^p)^{1/p}$ and $(A^p\sigma B)^{1/p}$ as $p\to\infty$ for positive matrices $A,B$, where $\Phi$ is a positive linear map between matrix algebras (in particular, $\Phi(A)=KAK^*$) and $\sigma$ is an operator mean (in particular, the weighted geometric mean), which are considered as certain reciprocal Lie-Trotter formulas and also a generalization of Kato's limit to the supremum $A\vee B$ with respect to the spectral order.Comment: 23 page

Quantum ff-divergences in von Neumann algebras II. Maximal ff-divergences

As a continuation of the paper [20] on standard $f$-divergences, we make a systematic study of maximal $f$-divergences in general von Neumann algebras. For maximal $f$-divergences, apart from their definition based on Haagerup's $L^1$-space, we present the general integral expression and the variational expression in terms of reverse tests. From these definition and expressions we prove important properties of maximal $f$-divergences, for instance, the monotonicity inequality, the joint convexity, the lower semicontinuity, and the martingale convergence. The inequality between the standard and the maximal $f$-divergences is also given.Comment: 38 page

A generalization of Araki's log-majorization

We generalize Araki's log-majorization to the log-convexity theorem for the eigenvalues of $\Phi(A^p)^{1/2}\Psi(B^p)\Phi(A^p)^{1/2}$ as a function of $p\ge0$, where $A,B$ are positive semidefinite matrices and $\Phi,\Psi$ are positive linear maps between matrix algebras. A similar generalization of the log-majorization of Ando-Hiai type is given as well.Comment: 16 pages, the last section is expande

Operator means deformed by a fixed point method

By means of a fixed point method we discuss the deformation of operator means and multivariate means of positive definite matrices/operators. It is shown that the deformation of an operator mean becomes again an operator mean. The means deformed by the weighted power means are particularly examined.Comment: 35 page

Operator kk-tone functions and analytic functional calculus

Operator $k$-tone functions on an open interval of the real line, which are higher order extensions of operator monotone and convex functions, are characterized via certain inequalities for the real and imaginary parts of analytic functional calculus by those functions.Comment: 19 page

Quantum ff-divergences in von Neumann algebras I. Standard ff-divergences

We make a systematic study of standard $f$-divergences in general von Neumann algebras. An important ingredient of our study is to extend Kosaki's variational expression of the relative entropy to an arbitary standard $f$-divergence, from which most of the important properties of standard $f$-divergences follow immediately. In a similar manner we give a comprehensive exposition on the R\'enyi divergence in von Neumann algebra. Some results on relative hamiltonians formerly studied by Araki and Donald are improved as a by-product.Comment: 33 page

Concavity of certain matrix trace and norm functions. II

We refine Epstein's method to prove joint concavity/convexity of matrix trace functions of Lieb type $\mathrm{Tr}\,f(\Phi(A^p)^{1/2}\Psi(B^q)\Phi(A^p)^{1/2})$ and symmetric (anti-) norm functions of the form $\|f(\Phi(A^p)\,\sigma\,\Psi(B^q))\|$, where $\Phi$ and $\Psi$ are positive linear maps, $\sigma$ is an operator mean, and $f(x^\gamma)$ with a certain power $\gamma$ is an operator monotone function on $(0,\infty)$. Moreover, the variational method of Carlen, Frank and Lieb is extended to general non-decreasing convex/concave functions on $(0,\infty)$ so that we prove joint concavity/convexity of more trace functions of Lieb type.Comment: 28 pages, a number of minor changes, Lemma A.3 adde

A log-Sobolev type inequality for free entropy of two projections

We prove an inequality between the free entropy and the mutual free Fisher information for two projections, regarded as a free analog of the logarithmic Sobolev inequality. The proof is based on the random matrix approximation procedure via the Grassmannian random matrix model of two projections.Comment: The assumption of the main theorem is improve

Conic structure of the non-negative operator convex functions on (0,∞)(0,\infty)

The conic structure of the convex cone of non-negative operator convex functions on $(0,\infty)$ (also on $(-1,1)$) is clarified. We completely determine the extreme rays, the closed faces, and the simplicial closed faces of this convex cone.Comment: 18 page