Sharp Quasi-Reverse Minkowski Inequality for Schatten Norms

Abstract: Let $|\cdot|_p$ denote the Schatten $p$-norm and let $|A|=(A<sup>*A)<sup>{1/2}$. For $2\leq p&lt;\infty$, let $x_p&gt;1$ be the unique solution of $x_p<sup>p=2x_p+1$, and set [ C_p=\frac{\sqrt{x_p(x_p+1)}}{(x_p^p+1)^{1/p}}. ] We prove the sharp inequality [ |A+B|_p\leq C_p\bigl||A|+|B|\bigr|_p ] for arbitrary complex matrices of arbitrary size. Equivalently, if $q=p/(p-1)$ and $R,X,Y$ are positive semidefinite, then [ |RX|_1+|RY|_1 \leq C_p|R|_q|X+Y|_p. ] For $1<p<2$, we also show that the formula proposed for the optimal constant fails. We give both a numerical counterexample and a systematic analytic construction.

• The paper proves that for every p≥2, the optimal constant is C_p=√[x_p(x_p+1)]/(x_p^p+1)^(1/p), where x_p^p=2x_p+1, with C_∞=√2 and rank-one examples showing sharpness.
• Its proof combines tensor domination, complex interpolation, and a rank-two reduction to control arbitrary matrices and derive an equivalent inequality for positive semidefinite operators.
• For every 1<p<2, the proposed formula fails—including an explicit p=3/2 counterexample—and higher-dimensional examples suggest that determining the true sharp constant may require dimension-dependent analysis.

Background and problem statement

For a complex matrix $A$ on a finite-dimensional Hilbert space $\mathcal H$, write $|A|=(A^*A)^{1/2}$ and let $\|\cdot\|_p$ denote the Schatten $p$-norm. The paper studies the optimal constant $c_p$ in the inequality

$\|A+B\|_p \le c_p \bigl\||A|+|B|\bigr\|_p,$

a "quasi-reverse Minkowski" estimate comparing the norm of a sum of operators with the norm of the sum of their moduli. The $p=2$ case was conjectured by Lee and proved by Lin–Zhang, with alternative proofs due to Zamani and Zhang. Tang and Zhang subsequently posed the optimal-constant problem for $m$ summands and proposed, for two summands, the explicit candidate

$$C_p=\frac{\sqrt{x_p(x_p+1)}}{(x_p^p+1)^{1/p}},\qquad x_p>1,\quad x_p^p=2x_p+1,$$

verifying it only at the endpoints $\mathcal H$0. The paper under review settles this conjecture affirmatively for all $\mathcal H$1, and refutes it for every $\mathcal H$2.

Main result

The central theorem states that for all $\mathcal H$3 and arbitrary matrices $\mathcal H$4,

$\mathcal H$5

with $\mathcal H$6 exactly as above, and that the constant is sharp; at $\mathcal H$7 the sharp constant is $\mathcal H$8. An equivalent dual formulation (Corollary on positive semidefinite triples) reads: for $\mathcal H$9 and positive semidefinite $|A|=(A^*A)^{1/2}$0,

$|A|=(A^*A)^{1/2}$1

The sharpness example is elementary: take rank-one operators $|A|=(A^*A)^{1/2}$2 and $|A|=(A^*A)^{1/2}$3 with $|A|=(A^*A)^{1/2}$4; the resulting quotient equals $|A|=(A^*A)^{1/2}$5 identically.

Proof architecture

The argument combines three independent ingredients.

Tensor domination lemma. For positive semidefinite $|A|=(A^*A)^{1/2}$6 with $|A|=(A^*A)^{1/2}$7, the paper proves

$|A|=(A^*A)^{1/2}$8

The proof is operator-algebraic: after compressing to the support of $|A|=(A^*A)^{1/2}$9, one forms Kraus operators $\|\cdot\|_p$0 satisfying $\|\cdot\|_p$1, so the associated completely positive trace-preserving map has spectral radius at most one. Its matrix representation $\|\cdot\|_p$2 is similar to the positive matrix $\|\cdot\|_p$3 with $\|\cdot\|_p$4, forcing all eigenvalues into $\|\cdot\|_p$5. A corollary extends this to two families $\|\cdot\|_p$6, $\|\cdot\|_p$7 simultaneously.

A Schatten-$\|\cdot\|_p$8 triangle-type inequality. Via the tensor result, the paper constructs a linear map $\|\cdot\|_p$9 that is contractive both from $p$0 to $p$1 and from $p$2 to $p$3. Complex interpolation between these noncommutative $p$4 spaces ($p$5, $p$6) yields

$p$7

Reduction to rank two. The key scalar computation concerns $p$8 positive semidefinite of rank at most two, with eigenvalues $p$9. Schatten Cauchy–Schwarz gives $c_p$0, and the ratio $c_p$1 is maximized precisely when $c_p$2, i.e., at $c_p$3, where its value is $c_p$4. Duality then converts this to $c_p$5 for rank-two $c_p$6.

The full theorem follows by decomposing $c_p$7 along its left singular vectors $c_p$8: each compression $c_p$9 is rank one, and writing $\|A+B\|_p \le c_p \bigl\||A|+|B|\bigr\|_p,$0, $\|A+B\|_p \le c_p \bigl\||A|+|B|\bigr\|_p,$1 with $\|A+B\|_p \le c_p \bigl\||A|+|B|\bigr\|_p,$2, the rank-two lemma bounds each singular value $\|A+B\|_p \le c_p \bigl\||A|+|B|\bigr\|_p,$3 by $\|A+B\|_p \le c_p \bigl\||A|+|B|\bigr\|_p,$4. Summing over $\|A+B\|_p \le c_p \bigl\||A|+|B|\bigr\|_p,$5 and applying the interpolated inequality with $\|A+B\|_p \le c_p \bigl\||A|+|B|\bigr\|_p,$6, $\|A+B\|_p \le c_p \bigl\||A|+|B|\bigr\|_p,$7 completes the proof. The structure is clean: pointwise rank-two control plus a global summation device handles arbitrary dimension without approximation arguments.

Failure of the proposed formula for $\|A+B\|_p \le c_p \bigl\||A|+|B|\bigr\|_p,$8

The paper shows the Tang–Zhang formula fails for every exponent in $\|A+B\|_p \le c_p \bigl\||A|+|B|\bigr\|_p,$9, first numerically and then analytically. At $p=2$0, the rank-one pair

$p=2$1

gives quotient $p=2$2, exceeding the proposed $p=2$3 — a strict counterexample.

The systematic construction considers $p=2$4, $p=2$5 with real unit vectors and inner products $p=2$6, $p=2$7. The quotient becomes an explicit function $p=2$8; for fixed $p=2$9 its unique maximizer over $m$0 is interior, attained at

$m$1

and choosing $m$2 with $m$3 yields $m$4. The mechanism is transparent: the Tang–Zhang extremizer corresponds to $m$5 (aligned left vectors), which is suboptimal once $m$6 because the exponent penalizes concentration differently.

Notably, even $m$7 is not the true sharp constant: at $m$8 a three-dimensional example achieves quotient $m$9 while $$C_p=\frac{\sqrt{x_p(x_p+1)}}{(x_p^p+1)^{1/p}},\qquad x_p>1,\quad x_p^p=2x_p+1,$$0. This suggests the universal sharp constant may not be attained by any fixed configuration and may grow with ambient dimension — an observation the paper records without resolving.

Limitations and open questions

Three caveats qualify the results. First, the main theorem covers only $$C_p=\frac{\sqrt{x_p(x_p+1)}}{(x_p^p+1)^{1/p}},\qquad x_p>1,\quad x_p^p=2x_p+1,$$1; the range $$C_p=\frac{\sqrt{x_p(x_p+1)}}{(x_p^p+1)^{1/p}},\qquad x_p>1,\quad x_p^p=2x_p+1,$$2 is left entirely open, with the sharp constant $$C_p=\frac{\sqrt{x_p(x_p+1)}}{(x_p^p+1)^{1/p}},\qquad x_p>1,\quad x_p^p=2x_p+1,$$3 in Question stated but not computed — the paper's own analysis shows neither the Tang–Zhang formula nor the two-parameter maximization of $$C_p=\frac{\sqrt{x_p(x_p+1)}}{(x_p^p+1)^{1/p}},\qquad x_p>1,\quad x_p^p=2x_p+1,$$4 provides it. Second, the multi-summand version of the Tang–Zhang conjecture for $$C_p=\frac{\sqrt{x_p(x_p+1)}}{(x_p^p+1)^{1/p}},\qquad x_p>1,\quad x_p^p=2x_p+1,$$5 remains untouched; the tensor lemma and interpolation step are formulated for general $$C_p=\frac{\sqrt{x_p(x_p+1)}}{(x_p^p+1)^{1/p}},\qquad x_p>1,\quad x_p^p=2x_p+1,$$6, but the rank-reduction argument is specific to two summands. Third, the possibility that $$C_p=\frac{\sqrt{x_p(x_p+1)}}{(x_p^p+1)^{1/p}},\qquad x_p>1,\quad x_p^p=2x_p+1,$$7 over configurations increases with dimension (raised by the $$C_p=\frac{\sqrt{x_p(x_p+1)}}{(x_p^p+1)^{1/p}},\qquad x_p>1,\quad x_p^p=2x_p+1,$$8 example) means finite-dimensional computations cannot certify sharp constants in low exponents without a dimension-independent argument.

Conclusion

The paper resolves the two-summand optimal-constant problem for Schatten norms in the range $$C_p=\frac{\sqrt{x_p(x_p+1)}}{(x_p^p+1)^{1/p}},\qquad x_p>1,\quad x_p^p=2x_p+1,$$9: the Tang–Zhang closed-form constant $\mathcal H$00, defined through the unique solution of $\mathcal H$01, is correct and sharp, proved via a tensor domination lemma, complex interpolation of noncommutative $\mathcal H$02 spaces, and exact rank-two reduction. In the complementary range $\mathcal H$03 the same formula provably fails for every exponent, and the true sharp constant is identified as a genuinely open problem whose solution likely requires going beyond rank-one extremizers.