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Eigenvalues inequalities involving (log) convex/concav functions and Hermitian matrices, positive unital maps are considered. Simple proofs of Bhatia-Kittaneh inequality and Naimark dilation theorem are given.
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This paper studies the relationship between vector-valued BMO functions and the Carleson measures defined by their gradients. Let $dA$ and $dm$ denote Lebesgue measures on the unit disc $D$ and the unit circle $mathbb T$, respectively. For $1< q<infty$ and a Banach space $B$ we prove that there exists a positive constant $c$ such that $$sup_{z_0in D}int_{D}(1-|z|)^{q-1}| abla f(z)|^q P_{z_0}(z) dA(z) le c^qsup_{z_0in D}int_{T}|f(z)-f(z_0)|^qP_{z_0}(z) dm(z)$$ holds for all trigonometric polynomials $f$ with coefficients in $B$ iff $B$ admits an equivalent norm which is $q$-uniformly convex, where $$P_{z_0}(z)=frac{1-|z_0|^2}{|1-bar{z_0}z|^2} .$$ The validity of the converse inequality is equivalent to the existence of an equivalent $q$-uniformly smooth norm.
Let $mathfrak{M}$ be a semifinite von Neumann algebra on a Hilbert space equipped with a faithful normal semifinite trace $tau$. A closed densely defined operator $x$ affiliated with $mathfrak{M}$ is called $tau$-measurable if there exists a number $lambda geq 0$ such that $tau left(e^{|x|}(lambda,infty)right)<infty$. A number of useful inequalities, which are known for the trace on Hilbert space operators, are extended to trace on $tau$-measurable operators. In particular, these inequalities imply Clarkson inequalities for $n$-tuples of $tau$-measurable operators. A general parallelogram law for $tau$-measurable operators are given as well.
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 $||| g(A)-g(B) ||| ge ||| g(|A-B|) |||$, obtained by Ando, for operator monotone $f$ with inverse $g$, also have a similar generalisation to non-negative concave $f$ and convex $g$. We answer exactly this question, in the negative for general matrices, and affirmatively in the special case when $Age ||B||$. In the course of this work, we introduce the novel notion of $Y$-dominated majorisation between the spectra of two Hermitian matrices, where $Y$ is itself a Hermitian matrix, and prove a certain property of this relation that allows to strengthen the results of Bourin-Uchiyama and Kosem, mentioned above.
We show norm estimates for the sum of independent random variables in noncommutative $L_p$-spaces for $1<p<infty$ following our previous work. These estimates generalize the classical Rosenthal inequality in the commutative case. Among applications, we derive an equivalence for the $p$-norm of the singular values of a random matrix with independent entries, and characterize those symmetric subspaces and unitary ideals which can be realized as subspaces of a noncommutative $L_p$ for $2<p<infty$.
A matrix convex set is a set of the form $mathcal{S} = cup_{ngeq 1}mathcal{S}_n$ (where each $mathcal{S}_n$ is a set of $d$-tuples of $n times n$ matrices) that is invariant under UCP maps from $M_n$ to $M_k$ and under formation of direct sums. We study the geometry of matrix convex sets and their relationship to completely positive maps and dilation theory. Key ingredients in our approach are polar duality in the sense of Effros and Winkler, matrix ranges in the sense of Arveson, and concrete constructions of scaled commuting normal dilation for tuples of self-adjoint operators, in the sense of Helton, Klep, McCullough and Schweighofer. Given two matrix convex sets $mathcal{S} = cup_{n geq 1} mathcal{S}_n,$ and $mathcal{T} = cup_{n geq 1} mathcal{T}_n$, we find geometric conditions on $mathcal{S}$ or on $mathcal{T}$, such that $mathcal{S}_1 subseteq mathcal{T}_1$ implies that $mathcal{S} subseteq Cmathcal{S}$ for some constant $C$. For instance, under various symmetry conditions on $mathcal{S}$, we can show that $C$ above can be chosen to equal $d$, the number of variables, and in some cases this is sharp. We also find an essentially unique self-dual matrix convex set $mathcal{D}$, the self-dual matrix ball, for which corresponding inclusion and dilation results hold with constant $C=sqrt{d}$. Our results have immediate implications to spectrahedral inclusion problems studied recently by Helton, Klep, McCullough and Schweighofer. Our constants do not depend on the ranks of the pencils determining the free spectrahedra in question, but rather on the number of variables $d$. There are also implications to the problem of existence of (unital) completely positive maps with prescribed values on a set of operators.
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