On an elementary operator with \(w\)-hyponormal operator entries
Let \(H\) be a complex Hilbert space. For a bounded linear operator \(A\) on \(H\), let \(A=U |A|\) be the polar decomposition. Let \(\tilde{A}=|A|^{1/2}U|A|^{1/2}\) and \(\tilde{A}^{*}=|A|^{1/2}U^{*}|A|^{1/2}\). Then \(A\) is said to be \(w\)-hyponormal if \(|\tilde{A}^{*}| \leq |A|\leq |\tilde{A}|.\) If \(A, B^*, C\), and \(D^*\) are \(w\)-hyponormal operators, \(A\) is doubly commuting with \(C\) and \(B\) is doubly commuting with \(D\), then \(AXB-CXD=0\) implies \(A^{*}XB^{*}-C^{*}XD^{*}=0\) for a bounded linear operator \(X\) on \(H\) if either \(B,C\) are invertible, \(A^{-1}(0) \subseteq A^{*-1}(0)\), and \(D^{-1}(0)\subseteq D^{*-1}(0)\), or \(C,D\) are invertible, \(A^{-1}(0) \subseteq A^{*-1}(0)\), and \(B^{*-1}(0)\subseteq B^{-1}(0)\). Hence the elementary operator \(X\mapsto AXB-CXD\) has the Putnam-Fuglede commutativity property. For bounded linear operators \(A\) and \(B\) on \(H\), let \(d_{AB}\) denote either generalized derivation \(X\mapsto AX-XB\) or the elementary operator \(X\mapsto AXB-X\). If \(A\) and \(B^*\) are \(w\)-hyponormal operators, then \(d_{AB}\) has the SVEP and the quasinilpotent part of \(d_{AB}-\lambda\) equals the kernel of \(d_{AB}-\lambda\) for every isolated eigenvalue \(\lambda\) of \(d_{AB}\). It follows that Weyl's theorem holds for \(d_{AB}\) and \(a\)-Weyl's theorem holds for its adjoint operator \(d_{AB}^{*}\). Actually, the last result of the paper says more. Namely, if \(A\) and \(B^*\) are \(w\)-hyponormal operators, \(f\) is a holomorphic function on \(\sigma(A)\) which is non-constant on every connected component of \(\sigma(A)\), and \(g\) is a holomorphic function on \(\sigma(B)\) which is non-constant on every connected component of \(\sigma(B)\), then, for every holomorphic function \(h\) on \(\sigma(d_{f(A)g(B)})\), Weyl's theorem holds for \(h(d_{f(A)g(B)})\) and \(a\)-Weyl's theorem holds for \(h(d_{f(A)g(B)}^{*})\).
scientific article; zbMATH DE number 5811024
scientific article