Is there a generalization of Araki-Lieb-Thirring inequality for four matrices?

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It is known that $$ Tr[(AB)^n] leq Tr (A^n B^n)$$ for $A$, $B$ positive semi-definite matrices. I am looking of a generalization of it for a product of $4$ positive semi-definite matrices of the the form,
$$Tr [(ABCD)^n] leq Tr [(AB)^2n ] Tr [(DC)^2n]$$
Is there any such inequality? Can someone suggest an approach if the above thing can be proved or disproved?




linear-algebra inequality trace positive-semidefinite




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    up vote
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    favorite












    It is known that $$ Tr[(AB)^n] leq Tr (A^n B^n)$$ for $A$, $B$ positive semi-definite matrices. I am looking of a generalization of it for a product of $4$ positive semi-definite matrices of the the form,
    $$Tr [(ABCD)^n] leq Tr [(AB)^2n ] Tr [(DC)^2n]$$
    Is there any such inequality? Can someone suggest an approach if the above thing can be proved or disproved?




    linear-algebra inequality trace positive-semidefinite




    share|cite|improve this question























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      It is known that $$ Tr[(AB)^n] leq Tr (A^n B^n)$$ for $A$, $B$ positive semi-definite matrices. I am looking of a generalization of it for a product of $4$ positive semi-definite matrices of the the form,
      $$Tr [(ABCD)^n] leq Tr [(AB)^2n ] Tr [(DC)^2n]$$
      Is there any such inequality? Can someone suggest an approach if the above thing can be proved or disproved?




      linear-algebra inequality trace positive-semidefinite




      share|cite|improve this question













      It is known that $$ Tr[(AB)^n] leq Tr (A^n B^n)$$ for $A$, $B$ positive semi-definite matrices. I am looking of a generalization of it for a product of $4$ positive semi-definite matrices of the the form,
      $$Tr [(ABCD)^n] leq Tr [(AB)^2n ] Tr [(DC)^2n]$$
      Is there any such inequality? Can someone suggest an approach if the above thing can be proved or disproved?





      linear-algebra inequality trace positive-semidefinite





      share|cite|improve this question












      share|cite|improve this question




      share|cite|improve this question








      edited Jul 28 at 17:05









      Daniel Buck

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      2,2841623









      asked Jul 28 at 15:46









      Jaswin

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