Małgorzata M. Czerwińska

arXiv2026

Towards a Characterization of Matrix Monotonicity via Radial Convexity

The theory of matrix monotone functions of finite degree depends subtly on the matrix size. Motivated by this phenomenon, we introduce radial convexity of metrics with respect to matrix orders on the cone of positive definite matrices. We establish radial convexity of several natural metrics with respect to the Loewner order and then focus on the Frobenius norm. Using the Daleckii-Krein formula, we relate radial convexity to the positivity of the Hadamard square of the corresponding Loewner matrix. This leads us to conjecture that radial convexity of order $n$ is equivalent to matrix monotonicity of degree $n$, and we prove this equivalence in degree $2$. We further provide evidence in higher dimensions, obtain a characterization in degree $3$, and discuss extensions of our results to arbitrary unitarily invariant norms.

math.FA↗

arXiv2015

k-Extreme Points in Symmetric Spaces of Measurable Operators

Let $\mathcal{M}$ be a semifinite von Neumann algebra with a faithful, normal, semifinite trace $τ$ and $E$ be a strongly symmetric Banach function space on $[0,τ(1))$. We show that an operator $x$ in the unit sphere of $E\left(\mathcal{M},τ\right)$ is $k$-extreme, $k\in\mathbb N$, whenever its singular value function $μ(x)$ is $k$-extreme and one of the following conditions hold (i) $μ(\infty,x)=\lim_{t\to\infty}μ(t,x)=0$ or (ii) $n(x)\mathcal{M} n(x^*)=0$ and $|x|\geq μ(\infty,x)s(x)$, where $n(x)$ and $s(x)$ are null and support projections of $x$, respectively. The converse is true whenever $\mathcal{M}$ is non-atomic. The global $k$-rotundity property follows, that is if $\mathcal{M}$ is non-atomic then $E$ is $k$-rotund if and only if $E\left(\mathcal{M},τ\right)$ is $k$-rotund. As a consequence of the noncommutive results we obtain that $f$ is a $k$-extreme point of the unit ball of the strongly symmetric function space $E$ if and only if its decreasing rearrangement $μ(f)$ is $k$-extreme and $|f|\geq μ(\infty,f)$. We conclude with the corollary on orbits $Ω(g)$ and $Ω'(g)$. We get that $f$ is a $k$-extreme point of the orbit $Ω(g)$, $g\in L_1+L_{\infty}$, or $Ω'(g)$, $g\in L_1[0,α)$, $α<\infty$, if and only if $μ(f)=μ(g)$ and $|f|\geq μ(\infty,f)$. From this we obtain a characterization of $k$-extreme points in Marcinkiewicz spaces.

math.FA↗