A condition for a subspace of \({\mathcal B} (H)\) to be an ideal. (English) Zbl 0852.46021

Summary: Let \(H\) be a real or complex Hilbert space with \(\dim H> 1\). The subspace \({\mathcal A}\subset {\mathcal B}(H)\) is an ideal if and only if \(TA- AT^*\in {\mathcal A}\) for every \(T\in {\mathcal B}(H)\), \(A\in {\mathcal A}\). Every element of \({\mathcal B}(H)\) is the finite sum of \(TA- AT^*\) type operators.


46B28 Spaces of operators; tensor products; approximation properties
47L10 Algebras of operators on Banach spaces and other topological linear spaces


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