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On an inequality of Lieb and Thirring

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Abstract

The following generalization of an inequality of Lieb and Thirring is proved:

$$Tr\{ b^{1 2} ab^{1 2} )^{qk} \} \leqslant Tr\{ (b^(q, 2) a^(q) b^(q 2)^k \} $$

for all positive selfadjoint operatorsa andb and for positive numbersq>1 andk>0. More generally,

$$Tr\varphi ((b^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} ab^{{1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} )q) \leqslant Tr\varphi (b^{qk} a^q b^{{q \mathord{\left/ {\vphantom {q 2}} \right. \kern-\nulldelimiterspace} 2}} q)$$

for any monotone increasing continuous function ϕ on (0, ∞) such that ϕ(0)=0 and ξ→ϕ(eξ) is convex.

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References

  1. GoldenS.,Phys. Rev. 137, B1127-B1128 (1965).

    Google Scholar 

  2. ThompsonC. J.,J. Math. Phys. 6, 1812–1813 (1965).

    Google Scholar 

  3. Lieb, E. and Thirring, W., in E. Lieb, B. Simon and A. S. Wightman (eds),Studies in Mathematical Physics, Princeton Press, 1976, pp. 301–302.

  4. WeylH.,Proc. Natl. Acad. Sci. U.S. 35, 408–411 (1949).

    Google Scholar 

  5. PolyaG.,Proc. Natl. Acad. Sci. U.S. 36, 49–51 (1950).

    Google Scholar 

  6. HeinzE.,Math. Ann. 123, 415–538 (1951).

    Google Scholar 

  7. Cordes, H. O.,Spectral Theory of Linear Differential Operators and Comparison Algebras, London Math. Soc. Lecture Note Series 76, Cambridge Univ. Press, 1987.

  8. Takayuki, Furuta, Equivalent norm inequalities to Löwner-Heinz theorem, to appear inRev. Math. Phys.

  9. FackT.,J. Operator Theory 7, 307–333 (1982).

    Google Scholar 

  10. FackT. and KosakiH.,Pacific J. Math. 123, 269–300 (1986).

    Google Scholar 

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Araki, H. On an inequality of Lieb and Thirring. Lett Math Phys 19, 167–170 (1990). https://doi.org/10.1007/BF01045887

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  • DOI: https://doi.org/10.1007/BF01045887

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