Fuglede-Putnam type theorems via the generalized Aluthge transform
Let \(H\) be a separable complex Hilbert space and \(L(H)\) be the algebra of all bounded linear operators on \(H\). For \(S, T\in L(H)\), let \(\mathrm{Com}(T,S)=\{ X\in L(H):\; TX=XS\}\). A pair \((T,S)\) is said to have the Fuglede-Putnam property if \(\mathrm{Com}(T,S) \subseteq \mathrm{Com}(T^*,S^*)\). Several results related to pairs with the Fuglede-Putnam property are presented. We mention a result which is close to the classical Fuglede-Putnam theorem. Theorem. If \(T\) is an invertible operator in the class \(A(s,t)\), where \(s+t=1\), and \(S^*\) is in the class \({\mathcal Y}\), then \(\mathrm{Com}(T,S) \subseteq \mathrm{Com}(T^*,S^*)\). Recall that \(T\in L(H)\) belongs to the class \(A(s,t)\) if \((|T^*|^t|T|^{2s}|T^*|^t)^{t/(t+s)} \geq |T^*|^{2t}\). If, for some \(\alpha\geq 1\), there exists a positive number \(k_\alpha\) such that \(|TT^*-T^*T|^\alpha\leq k_{\alpha}^{2}(T-\lambda)^*(T-\lambda)\) for all complex numbers \(\lambda\), then \(T\) is in the class \({\mathcal Y}\).
scientific article; zbMATH DE number 6341689
scientific article