Powers and commutativity of selfadjoint operators
Let \(B({\mathfrak H})\) be the algebra of bounded operators on a Hilbert space \(\mathfrak H\). An operator \(T\in B({\mathfrak H})\) is said to be accretive if \(\text{Re } T\geq 0\). It has been shown by \textit{C. R. DePrima} and \textit{B. K. Richard} [Indiana Univ. Math. J. 23, 163-172 (1973/74; Zbl 0257.47028)] that if \(T^n\) is accretive for \(n= 1,2,\dots\), then \(T\geq 0\). In the present paper this theorem is extended to pairs of operators, and some applications are given to proving commutativity of selfadjoint operators. For example, it is shown that if \(X,Y\in B({\mathfrak H})\), if \(\text{Re } X\geq \text{Re } Y\geq 0\), and if \(X^n+ Y^n\geq 0\) for \(n= 1, 2,\dots\), then \(X\geq Y\geq 0\). As an application of this result it is shown that if \(A\), \(B\) and \(C\) are selfadjoint operators for which \(B\geq C\), if \(AB^n+ C^n A\geq 0\) for \(n= 1, 2,\dots\), and if \(A\geq 0\) or \(C\geq 0\), then \(A\) commutes with \(B\) and \(C\).
scientific article; zbMATH DE number 734591