An extension of Uchiyama's result associated with an order preserving operator inequality
An extension of Uchiyama's result associated with an order preserving operator inequality
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内山结果的扩展与保序算子不等式相关
DOI：
10.7153/mia-05-14
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发表时间：
2002
期刊：
影响因子：
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通讯作者：
Eizaburo Kamei
中科院分区：
文献类型：
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作者：
T. Furuta;Eizaburo Kamei
令 A、B 和 C 为正可逆运算符，并令 r、s 和 t 为非负实数，使得 t s 和 (r, t) = (0, 0) 。那么下面的(I)和(II)成立并且相互推导。 (I) 如果对于所有 t 0 都有 At B∇λC （即 log At log(B∇λC )），则 f (t) = {A r 2 (B∇λC)A r 2 } s+r t+r 是 t 的增函数。 (II) 如果对于所有 t 0 都有 At B!λC（即 log At log(B!λC )），则 h(t) = {A 2 (B!λC)A r 2 } s+r t+r 是 t 的递减函数，其中 B∇λC 和 B!λC 分别是算术平均值和调和平均值。特别是，我们有 (I’) 如果在 B∇λC ，则 A r 2 (B∇λC)A r 2 {A 2 (B∇λC)A r 2 } s+r t+r (II’) 如果在 B!λC ，则 A r 2 (B!λC s)A r 2 {A r 2 (B!λC)A r 2 } s+r t+r 。这些是 Uchiyama [12] 最近结果的延伸。数学学科分类（2000）：47A63。
Let A,B and C be positive invertible operators and also let r, s and t be non-negative real numbers such that t s and (r, t) = (0, 0) . Then the following (I) and (II) hold and follows from each other. (I) If At B∇λC (i.e., log At log(B∇λC )) for all t 0 , then f (t) = {A r 2 (B∇λC)A r 2 } s+r t+r is an increasing function of t. (II) If At B!λC (i.e., log At log(B!λC )) for all t 0 , then h(t) = {A 2 (B!λC)A r 2 } s+r t+r is a decreasing function of t, where B∇λC and B!λC are the arithmetic mean and the harmonic mean respectively. In particular we have (I’) If At B∇λC , then A r 2 (B∇λC)A r 2 {A 2 (B∇λC)A r 2 } s+r t+r (II’) If At B!λC , then A r 2 (B!λC s)A r 2 {A r 2 (B!λC)A r 2 } s+r t+r . These are extensions of the recent results in Uchiyama [12]. Mathematics subject classification (2000): 47A63.