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Rodrigo de Azevedo
Rodrigo de Azevedo
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An extension of the Golden Thompson-Thompson inequality

For three symmetric positive-semidefinite semidefinite matrices $A,B,C$$A, B,C$, I am trying to figure out if the following inequality holds, at least in some cases:

$$tr(Ae^{B+C})≤tr(Ae^Be^C)$$ Note$$ \operatorname{tr} \left( A e^{B+C} \right) \leq \operatorname{tr} \left( A e^B e^C \right) $$

Note that if $A=I$ then this is the Golden Thompson inequalityGolden-Thompson inequality:

$$tr(e^{B+C})≤tr(e^Be^C)$$$$\operatorname{tr} \left( e^{B+C} \right) \leq \operatorname{tr} \left( e^B e^C \right)$$

I wonder if the first inequality is true or if adding certain assumptions (such as $A≼I$ i.e. $I−A$ is positive semidefinite) will suffice to prove it. I have tried searching for similar inequalities with no luck.

An extension of the Golden Thompson inequality

For three symmetric positive-semidefinite matrices $A,B,C$ I am trying to figure out if the following inequality holds, at least in some cases:

$$tr(Ae^{B+C})≤tr(Ae^Be^C)$$ Note that if $A=I$ then this is the Golden Thompson inequality:

$$tr(e^{B+C})≤tr(e^Be^C)$$

I wonder if the first inequality is true or if adding certain assumptions (such as $A≼I$ i.e. $I−A$ is positive semidefinite) will suffice to prove it. I have tried searching for similar inequalities with no luck.

An extension of the Golden-Thompson inequality

For three symmetric positive semidefinite matrices $A, B,C$, I am trying to figure out if the following inequality holds, at least in some cases:

$$ \operatorname{tr} \left( A e^{B+C} \right) \leq \operatorname{tr} \left( A e^B e^C \right) $$

Note that if $A=I$ then this is the Golden-Thompson inequality:

$$\operatorname{tr} \left( e^{B+C} \right) \leq \operatorname{tr} \left( e^B e^C \right)$$

I wonder if the first inequality is true or if adding certain assumptions (such as $A≼I$ i.e. $I−A$ is positive semidefinite) will suffice to prove it. I have tried searching for similar inequalities with no luck.

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nikka
nikka
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An extension of the Golden Thompson inequality

For three symmetric positive-semidefinite matrices $A,B,C$ I am trying to figure out if the following inequality holds, at least in some cases:

$$tr(Ae^{B+C})≤tr(Ae^Be^C)$$ Note that if $A=I$ then this is the Golden Thompson inequality:

$$tr(e^{B+C})≤tr(e^Be^C)$$

I wonder if the first inequality is true or if adding certain assumptions (such as $A≼I$ i.e. $I−A$ is positive semidefinite) will suffice to prove it. I have tried searching for similar inequalities with no luck.