Subspace gaps and Weyl's theorem for an elementary operator (Q2566598)

Author's abstract: A range-kernel orthogonality property is established for the elementary operators \({\mathcal E}(X)=\sum^n_{i=1}A_iXB_i\) and \({\mathcal E}_*(X)=\sum^n_{i=1}A_i^*XB_i^*\), where \({\mathbf A}=(A_1,A_2,\dots,A_n)\) and \({\mathbf B}=(B_1,B_2,\dots,B_n)\) are \(n\)-tuples of mutually commuting scalar operators (in the sense of Dunford) in the algebra \(B(H)\) of operators on a Hilbert space \(H\). It is proved that the operator \({\mathcal E}\) satisfies Weyl's theorem in the case in which \(\mathbf A\) and \(\mathbf B\) are \(n\)-tuples of mutually commuting generalized scalar operators.