JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2016, Vol. 51 ›› Issue (4): 19-24.doi: 10.6040/j.issn.1671-9352.0.2015.182

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Strongly and weakly separable operators on tensor product spaces

LIANG Wen-ting, CHEN Zheng-li*   

  1. School of Mathematics and Information Science, Shaanxi Normal University, Xian 710062, Shaanxi, China
  • Received:2015-04-23 Online:2016-04-20 Published:2016-04-08

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Abstract: The definitions of strongly separable and weakly separable operators are introduced. Operators of the form T=A⊗B are called separable operators. An operator T is said to be strongly separable if T maps all vectors to separable vectors; and a separable operator T is said to be weakly separable if Tx is separable implies x∈Cn is separable. Firstly, we give out a characterization of the rank of the corresponding components of vectors from Cn⊗Cn\{0} when the sum of those finite separable vectors is still separable. Then characterizations of a separable operator being strongly separable and weakly separable are obtaiend, and the necessary and sufficient conditions are given for the sum of two separable operators being strongly separable and weakly separable.

Key words: separable operator, strongly separable operator, weakly separable operator

CLC Number: 

  • O177.1
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