A remark on the range of elementary operators |
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Authors: | Bouali Said Bouhafsi Youssef |
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Affiliation: | 1.Rabat,Morocco;2.Kénitra,Morocco;3.Dept. of Math., Faculty of Science,Mohammed V Agdal University,Rabat,Morocco;4.Dept. of Math., Faculty of Science,Ibn Tofail University,Kénitra,Morocco |
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Abstract: | Let L(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Given A ∈ L(H), we define the elementary operator Δ A : L(H) → L(H) by Δ A (X) = AXA − X. In this paper we study the class of operators A ∈ L(H) which have the following property: ATA = T implies AT*A = T* for all trace class operators T ∈ C 1(H). Such operators are termed generalized quasi-adjoints. The main result is the equivalence between this character and the fact that the ultraweak closure of the range of Δ A is closed under taking adjoints. We give a characterization and some basic results concerning generalized quasi-adjoints operators. |
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