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Eun-Young Lee

Publications and source records attributed to Eun-Young Lee.

At least 19 recordsLinked to original sources
arXiv2026

Solutions to some open problems in matrix analysis

This paper deals with several open questions in matrix analysis. We answer a question of Audenaert and Kittaneh by strengthening the Aujla--Bourin subadditivity inequality. We also answer questions posed by Bourin and the second author: (1) we extend the reciprocal Lie--Trotter theorem of Audenaert and Hiai to a product involving a third matrix; (2) we determine the best constants in inequalities for sums of contractions and positive block matrices, including the triangle inequality $$ |A+B+C| \leq \frac{3}{4} I + |A|+|B|+|C| $$ for three contractions $A,B,C$, where $3/4$ cannot be replaced by a smaller constant; and (3) we prove an eigenvalue inequality involving the symmetric modulus $(|X|+|X^*|)/2$ studied by Bourin, Lee, and Zhang.

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arXiv2026

Some hybrid matrix triangle inequalities

A recent result due to Teng Zhang compares the sum of $m$ matrices and the sum of their quadratic symmetric moduli: $$ \left\| \sum_{k=1}^m A_k\right\| \le \sqrt{2} \left\| \sum_{k=1}^m |A_k|_{\qsym}\right\| $$ for every unitarily invariant norm. Here $|A|_{\qsym}$ is the quadratic mean of $|A|$ and $|A^*|$. We derive operator and eigenvalue refinements of Zhang's inequality from a new polar decomposition for the quadratic symmetric modulus. For instance, $$ \left| \sum_{k=1}^m A_k\right| \le \frac{\sqrt{2}}{2} \left\{ \sum_{k=1}^m \left(|A_k|_{\qsym}+V|A_k|_{\qsym}V^*\right)\right\} $$ for some unitary matrix $V$. We also establish the polar decomposition for the maximal modulus associated with Olson's order, and derive, as in the quadratic case, a series of estimates.

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arXiv2026

Averages over matrix unitary orbits and spectral order

We establish matrix versions of the comparisons between the $\ell^p$-norms or quasi-norms for sequences of complex numbers. For instance, given $1\ge q>0$, and a family of $m$ normal $d\times d$ matrices $A_1,\ldots, A_m$, we show that $$ \left|\sum_{k=1}^m A_k\right| \le \frac{1}{d}\sum_{i=1}^d V_i\left\{\sum_{k=1}^m |A_k|^{q}\right\}^{1/q}\!\!\!\!V_i^* $$ for some unitary $d\times d$ matrices $V_1,\ldots, V_d$. We also give applications to Olson's spectral order and to the comparison between the symmetric modulus and the quadratic symmetric modulus. In particular we show that the sum $A+B$ of two positive matrices submajorizes their Kato supremum $A\vee B$, thereby completing majorization results due to Ando.

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arXiv2026

Involutions and angles between subspaces

We provide a complete structure theorem for involutory matrices. This yields a new approach to principal angles between subspaces and provide a series of nice formulae for these angles.

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arXiv2020

Positive linear maps on normal matrices

For a positive linear map F and a normal matrix N, we show that |F(N)| is bounded by some simple linear combinations in the unitary orbit of F(|N|). Several elegant sharp inequalities are derived, especially for the Schur product.

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arXiv2020

Numerical range and positive block matrices

We obtain several norm and eigenvalue inequalities for positive matrices partitioned into four blocks. The results involve the numerical range of the off-diagonal block X, especially the distance from 0 to W(X).

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arXiv2016

Matrix inequalities from a two variables functional

Several matrix/operator inequalies are given. Most of them are unexpected extensions of the Araki Log-majorization theorem, obtained thanks to a new log-majorization for positive linear maps and normal operators (Theorem 2.9). The main idea and technical tool is a two variables log-convex norm functional (Theorem 1.2).

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arXiv2015

Pinchings and Positive linear maps

We employ the pinching theorem, ensuring that some operators A admit any sequence of contractions as an operator diagonal of A, to deduce/improve two recent theorems of Kennedy-Skoufranis and Loreaux-Weiss for conditional expectations onto a masa in the algebra of operators on a Hilbert space. We also get a few results for sums in a unitary orbit.

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arXiv2012

Decomposition and partial trace of positive matrices with Hermitian blocks

Let H be a positive semidefinite matrix partitioned into Hermitian blocks. Then, up to a direct sum operation, H is the average of matrices isometrically congruent to its partial trace. A few corollaries are given, related to important inequalities in quantum information theory such as the Nielsen-Kempe separability criterion.

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arXiv2012

Positive definite matrices with Hermitian blocks and their partial traces

Let $H$ be a positive semi-definite matrix partitioned in $β\times β$ Hermitian blocks, $H=[A_{s,t}]$, $1\le s,t,\le β$. Then, for all symmetric norms, {equation*} \| H \| \le \| \sum_{s=1}^β A_{s,s} \|. {equation*} The proof uses a nice decomposition for positive matrices and unitary congruences with the generators of a Clifford algebra. A few corollaries are given, in particular the partial trace operation increases norms of separable states on a real Hilbert space, leading to a conjecture for usual complex Hilbert spaces.

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