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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Said, B., Youssef, B. A remark on the range of elementary operators. Czech Math J 60, 1065–1074 (2010). https://doi.org/10.1007/s10587-010-0071-x
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DOI: https://doi.org/10.1007/s10587-010-0071-x