An Araki-Lieb-Thirring inequality for geometrically concave and geometrically convex functions

作者：Audenaert Koenraad M R^*

来源：Linear Algebra and Its Applications, 2013, 438(8): 3454-3462.

DOI：10.1016/j.laa.2013.01.001

• Log-majorisation
• Positive semidefinite matrix
• Matrix inequality

For positive definite matrices A and B, the Araki-Lieb-Thirring in-equality amounts to an eigenvalue log-submajorisation relation for fractional powers
lambda(A(t)B(t)) <(w(log)) lambda(t)(AB), 0 < t <= 1,
while for t >= 1, the reversed inequality holds. In this paper I generalise this inequality, replacing the fractional powers x(t) by a larger class of functions. I prove that a continuous, non-negative function f with domain dom(f) = [0, x(0)) for some positive x(0) (possibly infinity) satisfies
lambda(f(A)f(B)) <(w(log)) f(2)(lambda(1/2)(AB)),
for all positive semidefinite A and B with spectrum in dom(f), if and only if f is geometrically concave and 0 <= xf'(x) <= f(x) for all x is an element of dom(f). The reversed inequality holds for a continuous, non-negative function f if and only iff is geometrically convex and xf'(x) >= f(x) for all x is an element of dom(f). As an application I obtain a complementary inequality to the Golden-Thompson inequality.

• 出版日期2013-4-15