Counterexample to a question on commutators
Counterexample to a question on commutators
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关于换向器问题的反例
DOI：
10.1090/s0002-9939-1971-0276798-4
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发表时间：
1971
期刊：
影响因子：
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通讯作者：
A. Mcintosh
中科院分区：
文献类型：
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作者：
A. Mcintosh
本文证明了在具有有界交换子AB-BA的Hilbert空间H中，两个自伴算子A和B有可能具有如下性质：|.4|.B - B\A\是无界的（其中|A1表示A2的正平方根）。证明了对所有自然数n，存在一个有界正算子U和一个有界算子V满足\\UV-VU\\^n\UV+VU\\.导论.对上述问题的兴趣来自于如下事实：如果H = L 2（- oo，oo），Au =iu'且Bu = bu（其中&是a.e.可区别的函数），那么|^4JP -P| ^4|当A B-BA有界时（即当B'本质上有界时），注意|^4| P-P| ^4|是奇异积分算子（|AIE - P| f（x）= 7T_1 p.v. f（x-y）~2（B（x）-B（y））f（y）dy.这是Calderon（l）的一个更一般的定理的一维L2情形。他问T。加藤是否至少可以在抽象的环境中证明这种情况，特别是，是否|.4| P-P| ^4|凡此种种，皆是虚妄。虽然我们给出了两个AB-BA有界的算子，|J4| 5- B\A\无界，问题在于能否找到Calderon结果的抽象证明，特别是在特殊情况下（A =d/dx，B = B），AB-BA与B交换。所以知道下面问题的答案会很有趣：如果AB-BA是有界的，并且与B交换，|A \ B - B\A\必然有界？我们将在本文件的最后进一步评论这个问题。术语.如果A是Hilbert空间H中的线性算子，则D（A）表示A的整环。一个线性流形XED（A）称为A的一个核iiX在D（A）中在范数下是稠密的||«||;t =|| II|| 24-||;lw|| 2.在本文中，标量场被假定为复数C的域。结果。(I)存在两个线性算子A和B，
We show that it is possible for two selfadjoint opera- tors A and B in a Hilbert space H with bounded commutator AB—BA to have the property that |.4|.B — B\A\ is unbounded (where | A \ denotes the positive square root of A2). The proof re- duces to showing that for all natural numbers n, there exist a bounded positive operator U and a bounded operator V satisfying \\UV-VU\\^n\\UV+VU\\. Introduction. Interest in the above question arises from the fact that if H = L2( — oo, oo), Au =iu' and Bu = bu (where & is an a.e. differ- entiable function), then |^4JP —P|^4| is bounded whenever A B—BA is bounded (i.e. whenever b' is essentially bounded). Note that |^4|P—P|^4| is the singular integral operator (| AIE — P| ^41 )f(x) = 7T_1 p.v. f(x—y)~2(b(x)—b(y))f(y)dy. This is the one-dimensional L2 case of a more general theorem of Calderon (l). It was asked by T. Kato whether this case at least could be proved in an abstract setting, and in particular, whether |.4|P—P|^4| is bounded whenever AB—BA is bounded. Although we present two operators with AB—BA bounded and |j4|5— B\A\ unbounded, the question remains as to whether an abstract proof of Calderon's result can be found. In particular it is clear that in the special case (A =d/dx, B = b), AB — BA commutes with B. So it would be interesting to know the answer to the following question: If AB—BA is bounded and commutes with B, is | A \ B — B\A\ necessarily bounded? We comment further on this question at the end of the paper. Terminology. If A is a linear operator in a Hilbert space H, then D(A) denotes the domain of A. A linear manifold XED(A) is called a core oi A iiX is dense in D(A) under the norm ||«||;t = ||ii||24-||;lw||2. Throughout this paper the scalar field is assumed to be the field of complex numbers C. The result. (I) There exist two linear operators A and B in a