函数F(α)=(A~rB~αA~r)~((p+2r)/(α+2r))的最优单调区间

设p>0且A,B是Hilbert空间上两个正算子,Furuta给出若A>B>0,那么对任意r>0,F(α)=(ArBαAr)p+2r/α+2r 是关于α>P单调递减的,但是他指出这个结果在0<α0的条件下并不一定成立.本文给出: (1)如果-1/2-2r,那么F(α)在(-∞,-P-4r]及p,+∞)上单调递增,并且这两个区间不能扩大; (3)如果r>0,P>0,那么p,+∞)也是F(α)的最佳单调区间.

On the Best Monotonic Interval of the Function F(α) = (ArBαAr)p+2r/α+2r

Furuta showed that if A > B > 0, then for fixed p > 0 and r > 0, F(α) = (ArBαAr)p+2r/α+2r is decreasing for α > p. But he pointed out that this result does not remain valid for 0 < α < p and r > 0. In this paper, we show that (1) If -1/2 < r < 0 and p < -2r then F(α) is decreasing in -p - 4r,+∞) and (-∞,p], and the two intervals can not be enlarged; (2) If -1/2 < r < 0 and p > -2r then F(α) is increasing in (-∞, -p - 4r] and p,+∞), and the two intervals also can not be enlarged; (3) If r > 0, p > 0, then p, +∞) is also the best monotonic interval of F(α).