Some remarks on general commutators theorems (Q944125)

The paper under review is concerned with some generalizations of the classical theorem by Fuglede and Putnam about commutation of operators on Hilbert spaces. Let \(A\) be a unital complex Banach algebra and assume that \(a\in A\) is such that there exists \(b\in A\) with \(\max\{\|\exp(-\lambda b)\exp(\overline{\lambda} a)\|,\|\exp(-\overline{\lambda}a)\exp(\lambda b)\|\}= o(|\lambda|^{1/2})\) as \(|\lambda|\to\infty\). Among other results, it is shown that if \(x\in A\) is such that \(ax=xa\), then \(bx=xb\) and, if \(ax=xb\), then \(xa=bx\). The author also considers some asymptotical versions of the Fuglede--Putnam theorem for some operators acting on complex Banach spaces.