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Let $H_n$ denote the space of all hermitian $n$ by $n$ matrices. Let $\mathbb{R}[H_n]$ denote the space of all real-valued polynomials $f$ with real coefficients in the entries of a variable matrix $A \in H_n$. Further, denote by $\mathbb{R}[H_n]^G$ the space of all such polynomials $f$, which are invariant under the adjoint action of $G = U(n)$ on $H_n$. In other words, it consists of polynomials $f \in \mathbb{R}[H_n]$ such that $f(gAg^{-1}) = f(A)$ for any unitary $n$ by $n$ matrix $g$.

Note that this is pretty much the same setting as in the Chern-Weil homomorphism (for the case where $G$ is the unitary group), except we are working mostly over $\mathbb{R}$ as the ground field.

Now let $f \in \mathbb{R}[H_n]^G$ be homogeneous of degree $k$. Denote also by $f$ its complete polarization, so that $f$ is a symmetric multilinear form depending of $k$ hermitian matrices $A_1, \ldots, A_k$.

Assume that $f(A_1,\ldots,A_k) \geq 0$ in the case where the $A_i$ are hermitian positive semidefinite and pairwise commute with each other. Does the inequality then hold for all hermitian positive semidefinite matrices $A_1, \ldots, A_k$? Or is there some counterexample to this "principle", by which I mean that there is some $f$ such as above such that the inequality $f(A_1,\ldots,A_k) \geq 0$ holds for hermitian positive semidefinite matrices $A_1, \ldots, A_k$ which pairwise commute with each other, but not for all $k$-tuples of positive semidefinite hermitian matrices?

I think it is a reasonable thing to ask.

Malkoun
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    Does this answer your question? What are the conditions on $\text{tr}(AB) \leq \text{tr(A)} \text{tr(B)}$ to be true? – user8675309 Apr 11 '21 at 17:46
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    @user8675309, it does answer the first part of my post, which I have now deleted. Thank you! But then I asked a much more general question, which may or may not be true, and I am interested in that more general question too. So I erased the first part of my post, which was indeed a duplicate as you have noticed, and left the more general question, which is interesting, even if it is probably too good to be true, and is thus probably false... Still, it is worth investigating. – Malkoun Apr 11 '21 at 18:46
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    your first question was quite simple and well defined. Your second (and now sole) question is not. One really basic problem: what does $f(A_1,\ldots,A_{\kappa}) \geq 0$ even mean? – user8675309 Apr 11 '21 at 20:38
  • @user8675309, sorry I was away from my laptop. I have just edited my post and explained in very precise terms what I mean. – Malkoun Apr 11 '21 at 23:20
  • It looks like you are trying to make a ring here but the product of hermitian matrices need not be hermitian which makes statements like "polynomials $f$ with real coefficients in the entries of a variable matrix $A \in H_n$" not directly parse-able. Anyway I guessed what you were after and what I would have suggested is $\det\big(A_1A_2 +A_2A_1\big)$ – user8675309 Apr 12 '21 at 00:38
  • @user8675309, yes you are right. A valid point. – Malkoun Apr 12 '21 at 00:52

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I found a counterexample... For example, one may consider

$$\DeclareMathOperator{\tr}{tr} f(A_1,A_2,A_3) = \tr(A_1 A_2 A_3) + \tr(A_1 A_3 A_2)$$

with $n = 2$ (i.e. the $A_i$ are hermitian $2$ by $2$ matrices). Then by randomly generating $1000$ random triples of hermitian positive definite matrices, I found that for that sample, $977$ satisfy the inequality $f(A_1,A_2,A_3) \geq 0$ but the other $23$ do not... So there goes that idea...

Here is a concrete/explicit counterexample.

$$A_1 = \left( \begin{array}{cc} 10 & -4 + 5i \\ -4 - 5i & 5 \end{array} \right),$$

$$A_1 = \left( \begin{array}{cc} 4 & -1 - 3i \\ -1 + 3i & 4 \end{array} \right),$$

$$A_1 = \left( \begin{array}{cc} 2 & 3 - 0.5i \\ 3 + 0.5i & 5 \end{array} \right),$$

then $A_1$, $A_2$ and $A_3$ are positive definite hermitian matrices but $f(A_1,A_2,A_3) = -71 < 0$.

Malkoun
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