The following generalization of an inequality of Lieb and Thirring is proved:
ab^1 2 )^qk Tr (b^(q, 2) a^(q) b^(q 2)^k Tr\{ b^{1 2} ab^{1 2} )^{qk} \} \leqslant Tr\{ (b^(q, 2) a^(q) b^(q 2)^k \}
|
for all positive selfadjoint operators
a and
b and for positive numbers
q>1 and
k>0. More generally,
Trj((b1 \mathord | / |
\vphantom 1 2 2 ab1 \mathord | / |
\vphantom 1 2 2 )q) \leqslant Trj(bqk aq bq \mathord | / |
\vphantom q 2 2 q)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
on (0,
) such that
(0)=0 and
(e
) is convex.
AMS subject classification (1980) 47B10