arXiv subjects

Mario Krnic

Publications and source records attributed to Mario Krnic.

4 recordsLinked to original sources

arXiv2025

Matrix-Test Duality: A Support-Function Characterization for $C^*$-Convex Families of CP Maps

We develop a matrix-test dual framework for $C^*$-convex families of completely positive maps $\CP(\mathscr S,\mathscr T)$, where $\mathscr S$ is an operator system and $\mathscr T$ is a unital $C^*$-algebra. Matrix tests $(k,f,s)$ induce evaluation functionals $\Phi\mapsto f(\Phi_k(s))$ and generate a natural weak topology $\tau=\sigma(\mathcal E,\mathcal F)$ on $\mathcal E=\mathrm{span}_{\mathbb C}(\CP(\mathscr S,\mathscr T))$. Our main result provides a support-function/separation characterization of the $\tau$-closed $C^*$-convex hull $\overline{\cconv(\mathcal K)}^{\,\tau}$ of a family $\mathcal K\subseteq \CP(\mathscr S,\mathscr T)$ in terms of matrix-test inequalities. A key technical tool is a finite-dimensional folding procedure that compresses finite linear combinations of test functionals into a single higher-level matrix test. As consequences, we obtain a single-test witness for non-membership, support-function criteria for inclusion and equality of $\tau$-closed $C^*$-convex hulls, and, under $0\in\overline{\cconv(\mathcal K)}^{\,\tau}$, an exact normalized bipolar-type reconstruction statement. We also show that $\tau$ is already generated by level-$1$ tests, although higher matrix levels remain essential in the geometric test inequalities.

math.OA↗

arXiv2020

Bounds for the $p$-angular distance and characterizations of inner product spaces

Based on a suitable improvement of a triangle inequality, we derive new mutual bounds for $p$-angular distance $α_p[x,y]=\big\Vert \Vert x\Vert^{p-1}x- \Vert y\Vert^{p-1}y\big\Vert$, in a normed linear space $X$. We show that our estimates are more accurate than the previously known upper bounds established by Dragomir, Hile and Maligranda. Next, we give several characterizations of inner product spaces with regard to the $p$-angular distance. In particular, we prove that if $|p|\geq |q|$, $p\neq q$, then $X$ is an inner product space if and only if for every $x,y\in X\setminus \{0\}$, $${α_p[x,y]}\geq \frac{{\|x\|^{p}+\|y\|^{p} }}{\|x\|^{q}+\|y\|^{q} }α_q[x,y].$$

math.FA↗

arXiv2018

Interpolating operator Jensen-type inequalities for log-convex and superquadratic functions

Motivated by some recently established operator Jensen-type inequalities related to a usual convexity, in the present paper we derive several more accurate operator Jensen-type inequalities for certain subclasses of convex functions. More precisely, we obtain interpolating series of Jensen-type inequalities for log-convex and non-negative superquadratic functions. In particular, we obtain the corresponding refinements of the Jensen-Mercer operator inequality for such classes of functions.

math.FA↗

arXiv2014

Reverses of the Young inequality for matrices and operators

We present some reverse Young-type inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with operator means. More precisely, we show that if $A, B\in {\mathfrak B}(\mathcal{H})$ are positive operators and $r\geq 0$, $A\nabla_{-r}B+2r(A\nabla B-A\sharp B)\leq A\sharp_{-r}B$ and prove that equality holds if and only if $A=B$. We also establish several reverse Young-type inequalities involving trace, determinant and singular values. In particular, we show that if $A, B$ are positive definite matrices and $r\geq 0$, then $\label{reverse_trace} \mathrm{tr}((1+r)A-rB)\leq \mathrm{tr}|A^{1+r}B^{-r} |-r(\sqrt{\mathrm{tr} A} - \sqrt{\mathrm{tr} B})^{2}$.

math.FA↗