Absolutely norm attaining paranormal operators
Absolutely norm attaining paranormal operators
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绝对正常达到超自然操作员
DOI：
10.1016/j.jmaa.2018.05.024
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发表时间：
2018
影响因子：
1.3
通讯作者：
G. Ramesh
中科院分区：
文献类型：
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作者：
G. Ramesh
有界线性算子T：H1 → H2，其中H1，H2是Hilbert空间，如果存在一个单位向量x∈ H1使得<$Tx <$=<$Tx <$.如果对H1的任意闭子空间M，|M：T的M→ H2到M是赋范算子，则T称为绝对赋范算子或AN-算子。本文证明了如下特征定理：定义在无穷维Hilbert空间H上的正算子T是AN-算子当且仅当T的本质谱是单点且[m（T），me（T））至多包含n个点.这里m（T）和me（T）分别是T的最小模和本质最小模。作为结果，我们得到了一个充分条件，在该条件下，算子的AN-性质蕴涵其伴随的AN-性质。研究了超正规AN-算子的结构，给出了超正规AN-算子正规的一个充要条件。
A bounded linear operator T: H 1→ H 2, where H 1, H 2 are Hilbert spaces is said to be norm attaining if there exists a unit vector x∈ H 1 such that‖ T x‖=‖ T‖. If for any closed subspace M of H 1, the restriction T| M: M→ H 2 of T to M is norm attaining, then T is called an absolutely norm attaining operator or AN-operator. We prove the following characterization theorem: a positive operator T defined on an infinite dimensional Hilbert space H is an AN-operator if and only if the essential spectrum of T is a single point and [m (T), m e (T)) contains atmost finitely many points. Here m (T) and m e (T) are the minimum modulus and essential minimum modulus of T, respectively. As a consequence we obtain a sufficient condition under which the AN-property of an operator implies AN-property of its adjoint. We also study the structure of paranormal AN-operators and give a necessary and sufficient condition under which a paranormal AN-operator is normal.