On weakly -differentiable operators

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• 作者：Erik Christensen ^{echris@math.ku.dk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：37A55 ; 46L55 ; 47D06 ; 58B34 ; 81S05
• 刊名：Expositiones Mathematicae
• 出版年：2016
• 出版时间：2016
• 年：2016
• 卷：34
• 期：1
• 页码：27-42
• 全文大小：223 K

文摘

Let D$D$ be a self-adjoint operator on a Hilbert space H$H$ and a$a$ a bounded operator on H$H$. We say that a$a$ is weakly D$D$-differentiable, if for any pair of vectors ξ,η$\xi ,\eta$ from H$H$ the function 〈e^itDae^−itDξ,η〉$〈e^{itD}a{e}^{-itD}\xi ,\eta 〉$ is differentiable. We give an elementary example of a bounded operator a$a$, such that a$a$ is weakly D$D$-differentiable, but the function e^itDae^−itD$e^{itD}a{e}^{-itD}$ is not uniformly differentiable. We show that weak  D$D$-differentiability   may be characterized by several other properties, some of which are related to the commutator (Da−aD)$(Da-aD)$.