bibkey: hayajneh2018uinorm
authors: Mostafa Hayajneh; Saja Hayajneh; Fuad Kittaneh
year: 2018
title: On some classical trace inequalities and a new Hilbert–Schmidt norm inequality
doi: 10.7153/mia-2018-21-81
url: https://doi.org/10.7153/mia-2018-21-81
claim: Conjecture 1 proposes monotonicity of the mixed-power expression in every unitarily invariant norm for positive semidefinite A and Hermitian B.
strata_touched:
  - D5/S3/Quantum/Schatten/TraceNorm
  - D5/S3/Quantum/Matrix/HayajnehConjectureRefutation
license: citation-only
triage: anchor

On some classical trace inequalities and a new Hilbert–Schmidt norm inequality

Locator

• Journal: Mathematical Inequalities & Applications 21(4) (2018), 1175–1183.
• DOI: https://doi.org/10.7153/mia-2018-21-81.
• Journal PDF: https://files.ele-math.com/articles/mia-21-81.pdf.
• Location: §1, Conjecture 1, printed p. 1177.
• The pinned source is the journal version. No arXiv identifier is asserted.

Source statement and definitions

CONJECTURE 1. Let A be a positive semidefinite matrix and B be a Hermitian matrix. Then for 1/2 ⩽ s ⩽ t ⩽ 1 and for every unitarily invariant norm, we have |||A^s B + BA^{1−s}||| ⩽ |||A^t B + BA^{1−t}|||.

The statement quantifies over every finite dimension, all complex positive semidefinite matrices $A$, all complex Hermitian matrices $B$, all real parameters $1/2\le s\le t\le 1$, and every unitarily invariant norm on that matrix space. Real powers use continuous functional calculus. The conjecture imposes no requirement $A\ge I$ and no spectral-range restriction.

Counterexample

Take

$$A=\left(\begin{array}{cc}1/81& 0\\ 0& 1\end{array}\right),B=\left(\begin{array}{cc}1& 3/4\\ 3/4& 1\end{array}\right),s=\frac{1}{2},t=\frac{3}{4}.$$

Both matrices are positive definite: their eigenvalues are $1/81,1$ and $1/4,7/4$, respectively. In particular, the required positive semidefiniteness and Hermitian hypotheses hold. Functional calculus on the diagonal matrix gives

$$A^{1/2}=\mathrm{diag}(1/9,1),A^{1/4}=\mathrm{diag}(1/3,1),A^{3/4}=\mathrm{diag}(1/27,1).$$

Consequently the two compared matrices are

$$L=A^{1/2}B+B{A}^{1/2}=\left(\begin{array}{cc}2/9& 5/6\\ 5/6& 2\end{array}\right),R=A^{3/4}B+B{A}^{1/4}=\left(\begin{array}{cc}10/27& 7/9\\ 1& 2\end{array}\right).$$

For every complex two by two matrix $T$, the trace norm satisfies

$$\Vert T{\Vert }_1^2=\mathrm{tr}(T^{\ast }T)+2|\det T|.$$

Indeed, if its singular values are ${\sigma }_1,{\sigma }_2\ge 0$, then ${\sigma }_1^2+{\sigma }_2^2=\mathrm{tr}(T^{\ast }T)$ and ${\sigma }_1{\sigma }_2=|\det T|$. This identity applies to $R$ even though $R$ is not Hermitian. Here

$$\det L=-\frac{1}{4},\det R=-\frac{1}{27},\Vert L{\Vert }_1^2=\frac{481}{81},\Vert R{\Vert }_1^2=\frac{4240}{729}.$$

Thus

$$\Vert L{\Vert }_1^2-\Vert R{\Vert }_1^2=\frac{89}{729}>0,\Vert L{\Vert }_1>\Vert R{\Vert }_1.$$

The trace norm is unitarily invariant, so this strict reverse comparison refutes Conjecture 1. It also violates the proposed singular-value weak majorization at its full prefix $k=2$, the sum of both singular values.

Scope and literature boundary

The conjecture and the known positive cases are attributed to Hayajneh, Hayajneh and Kittaneh. The explicit rational matrices and exact trace-norm comparison are repository derivations. The source proves the Hilbert–Schmidt case in §2 and the endpoint case $s=1/2,t=1$ for all unitarily invariant norms on p. 1179: “This affirmatively settles Conjecture 1 for the case s = 1/2 and t = 1.” Neither case is contradicted by the example.

The literature check covered the DOI and title with conjecture/counterexample/proof; “Hayajneh” “Conjecture 1”; and A^s B + BA^{1−s} with unitarily invariant norm. Related work checked: S. Hayajneh, Filomat 38(19) (2024), 6891–6896, which reproves a related scalar trace inequality; Bottazzi–Elencwajg–Larotonda–Varela, arXiv:1403.7472, which concerns a different expression, A^tB^{1−t} + B^tA^{1−t}. No settlement of Conjecture 1 was found within that scope. Citation indexes were not exhaustively checked.

For the displayed pair, numerical evaluation gives the conjectured direction for Schatten $p\in \{1.5,2,3,\infty \}$. This finite list does not establish the inequality for all Schatten $p\ge 2$ or for the operator norm on arbitrary inputs. Those restricted universal questions remain open here. Absence of later settlements and worldwide priority are not established. This citation-only note incorporates no external source software.