Some operator monotone functions
Some operator monotone functions
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DOI：
10.1090/s0002-9939-1972-0306957-4
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发表时间：
1972
期刊：
影响因子：
--
通讯作者：
G. Pedersen
中科院分区：
文献类型：
--
作者：
G. Pedersen
利用C*-代数理论给出了著名定理的一个简短证明：如果S和Tar是Hubert空间上的有界自伴算子，使得0^5^T，则Sai| 7“a对于每个0 ^a 0;并且S+e/和T+el都是可逆的。由于（S+eI）x在e-0对每个a>0时依范数收敛于Sx，并且由于B（%>）中的正算子形成一个范数闭集，所以假设S和T是可逆的就足以证明定理。(The a=0的情况可以直接验证，因为S°是5的距离投影。）设E表示[0，1]中的指数a的集合，其中函数t-t”是算子单调的。由于函数a-S”在范数拓扑中从[0，1]到F（§）是连续的，我们看到F是闭集。当我们证明F是凸的时，证明就完成了。取E中的a和s。则S°^TX;因此r^^F”“72^/。由此得出，类似地\\Ss'2 T-s/2\\^\。设P（A）为算子A的谱半径，我们有p（AB）= p（BA）。因此，/j*-U+s）/lMa+s）/2j--<i>+s1li）_ D（'r'.a-s）li'j-iix+s'iHc（a+s）lij-l.！i+s）H-Y-t.！ii-s）li_n（T-f/！c（»+j！）/ 2T-a/2 <; II j-0/2c（<t+/>/2<T-n/2t| <; \\T-pl2Ssl2\\||S°，/2T_，I/21| 1.编辑于1972年4月27日收到。AMS 1970主题分类。小学47 B15;中学46 L05。
A short proof is given based on C*-algebra theory for the well-known theorem that if S and Tare bounded selfadjoint operators on a Hubert space such that 0^5^T then Sai|7"a for each0^a 0; and S+e/and T+el are both invertible. Since (S+eI)x converges to Sx in norm when e-0 for each a>0, and since the positive operators in B(%>) form a norm closed set, it suffices to prove the theorem assuming that S and T are invertible. (The case a=0 can be verified directly, since S° is the range projection of 5.) Let E denote the set of exponents a in [0, 1] for which the function t—t" is operator monotone. Trivially 0e£ and 1 eE. Since the function a-S" is continuous from [0, 1] to F(§) in the norm topology we see that Fis a closed set. The proof will be complete when we show that Fis convex. Take a and s in E. Then S°^TX; hence r^^F""72^/. It follows that \\S"/2T-"'*\\£l. Similarly \\Ss'2T-s/2\\^\. With P(A) the spectral radius of an operator A we have p(AB) = p(BA). Therefore /j*-U+s)/lMa+s)/2j--<"+s1li) _ D('r'.a-s)li'j—iix+s'iHc{a+s)lij-l.!i+s)H-Y-t.!ii-s)li\ _ „(T—f/!c(»+j!)/2T-a/2\ <; II j—0/2c(«t+/»/2<T--«/2t| <; \\T-pl2Ssl2\\ ||S°,/2T_,I/21| 1. Received by the editors April 27, 1972. AMS 1970 subject classifications. Primary 47B15; Secondary 46L05.