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In this paper, we introduce the α-z-Bures Wasserstein divergence for positive semidefinite matrices A and B as Φ ( A , B ) = T r ( ( 1 − α ) A + α B ) − T r ( Q α , z ( A , B ) ) , where Q α , z ( A , B ) = ( A 1 − α 2 z B α z A 1 − α 2 z ) z is the matrix function in the α-z-Renyi relative entropy. We show that for 0 ≤ α ≤ z ≤ 1 , the quantity Φ ( A , B ) is a quantum divergence and satisfies the Data Processing Inequality in quantum information. We also solve the least squares problem with respect to the new divergence. In addition, we show that the matrix power mean μ ( t , A , B ) = ( ( 1 − t ) A p + t B p ) 1 / p satisfies the in-betweenness property with respect to the α-z-Bures Wasserstein divergence.

In this paper we establish some Positivstellensatze for polynomial matrices, applying the Scherer–Hol theorem. Firstly, we give a representation for polynomial matrices positive definite on subsets of compact polyhedra. Then we establish a Putinar-Vasilescu Positivstellensatz for polynomial matrices. Next we propose a matrix version of the Dickinson–Povh Positivstellensatz. Finally, we establish a version of Marshall’s theorem for polynomial matrices, approximating positive semi-definite polynomial matrices using sums of squares.

Let $$\sigma$$ be a non-trivial operator mean in the sense of Kubo and Ando, and let $$OM_+^1$$ be the set of normalized positive operator monotone functions on $$(0, \infty )$$ . In this paper, we study the class of $$\sigma$$ -subpreserving functions $$f\in OM_+^1$$ satisfying $$\begin{aligned} f(A\sigma B) \le f(A)\sigma f(B) \end{aligned}$$ for all invertible positive operators A and B. We provide some criteria for f to be trivial, i.e., $$f(t)=1$$ or $$f(t)=t$$ . We also establish characterizations of $$\sigma$$ -preserving functions $$f\in OM_+^1$$ satisfying $$\begin{aligned} f(A\sigma B) = f(A)\sigma f(B) \end{aligned}$$ for all invertible positive operators A and B. In particular, when $$\lim _{t\rightarrow 0} (1\sigma t) =0$$ , the function $$f\in OM_+^1\backslash \{1,t\}$$ preserves $$\sigma$$ if and only if f and $$1\sigma t$$ are representing functions for a weighted harmonic mean.

Bhatia, Lim, and Yamazaki studied the norm minimality of several Kubo-Ando means of positive semidefinite matrices. Recently, Hiai proved a norm minimality result involving the the weighted geometr...

In this paper we study the monotonicity, in-betweenness and in-sphere properties of matrix means with respect to Bures–Wasserstein, Hellinger and log-determinant metrics. More precisely, we show that the matrix power means (Kubo–Ando and non-Kubo–Ando extensions) satisfy the in-betweenness property in the Hellinger metric. We also show that for two positive definite matrices A and B, the curve of weighted Heron means, the geodesic curve of the arithmetic and the geometric mean lie inside the sphere centered at the geometric mean with the radius equal to half of the log-determinant distance between A and B.

We extend inequalities for operator monotone and operator convex functions onto elements of the extended positive part of a von Neumann algebra. In particular, this provides an opportunity to extend the inequalities onto unbounded positive self-adjoint operators.

The in-sphere property for matrix means is studied. It is proved that the matrix power mean satisfies in-sphere property with respect to the Hilbert-Schmidt norm. A new characterization of the matrix arithmetic mean is provided. Some reverse AGM inequalities involving unitarily invariant norms and operator monotone functions are also obtained.

Bhatia, Lim, and Yamazaki conjectured that the Kubo–Ando extensions of means of numbers satisfy a norm minimality condition with respect to unitarily invariant norms. In this short note, we introduce a symmetric Kubo–Ando mean and a non-Kubo–Ando extension that do not satisfy this property.

In this paper, via Newton polyhedra, we define and study symmetric matrix polynomials which are nondegenerate at infinity. From this, we construct a class of (not necessarily compact) semialgebraic sets in $\mathbb {R}^{n}$ such that for each set K in the class, we have the following two statements: (i) the space of symmetric matrix polynomials, whose eigenvalues are bounded on K, is described in terms of the Newton polyhedron corresponding to the generators of K (i.e., the matrix polynomials used to define K) and is generated by a finite set of matrix monomials; and (ii) a matrix version of Schmudgen’s Positivstellensatz holds: every matrix polynomial, whose eigenvalues are “strictly” positive and bounded on K, is contained in the preordering generated by the generators of K.

In this short note we prove a conjecture due to Bhatia, Lim, and Yamazaki on the matrix power means. As a consequence, we obtain a related inequality for the matrix Heron mean and its non-Kubo–Ando extension, conjectured by Bhatia et al..

In this article, we provide an alternate proof of the fact that the weighted power means μ p ( A , B , t ) = ( t A p + ( 1 − t ) B p ) 1 / p , 1 ≤ p ≤ 2 satisfy Audenaert's “in-betweenness” property for positive semidefinite matrices. We show that the “in-betweenness” property holds with respect to any unitarily invariant norm for p = 1 / 2 and with respect to the Euclidean metric for p = 1 / 4 . We also show that the only Kubo–Ando symmetric mean that satisfies the “in-betweenness” property with respect to any metric induced by a unitarily invariant norm is the arithmetic mean. Finally, for p = 6 we give a counterexample to a conjecture by Audenaert regarding the “in-betweenness” property.

Let $p$ be a positive number and $h$ a function on $\mathbb{R}^+$ satisfying $h(xy) \ge h(x) h(y)$ for any $x, y \in \mathbb{R}^+$. A non-negative continuous function $f$ on $K (\subset \mathbb{R}^+)$ is said to be {\it operator $(p,h)$-convex} if \begin{equation*}\label{def} f ([\alpha A^p + (1-\alpha)B^p]^{1/p}) \leq h(\alpha)f(A) +h(1-\alpha)f(B) \end{equation*} holds for all positive semidefinite matrices $A, B$ of order $n$ with spectra in $K$, and for any $\alpha \in (0,1)$. In this paper, we study properties of operator $(p,h)$-convex functions and prove the Jensen, Hansen-Pedersen type inequalities for them. We also give some equivalent conditions for a function to become an operator $(p,h)$-convex. In applications, we obtain Choi-Davis-Jensen type inequality for operator $(p,h)$-convex functions and a relation between operator $(p,h)$-convex functions with operator monotone functions.

We study some geometric properties associated with the t-geometric means A ♯ t B:= A 1/2(A −1/2 BA −1/2) t A 1/2 of two n × n positive definite matrices A and B. Some geodesical convexity results with respect to the Riemannian structure of the n × n positive definite matrices are obtained. Several norm inequalities with geometric mean are obtained. In particular, we generalize a recent result of Audenaert (2015). Numerical counterexamples are given for some inequality questions. A conjecture on the geometric mean inequality regarding m pairs of positive definite matrices is posted.