On some product of two unbounded self-adjoint operators (Q1035224)

The Rehder-Albrecht-Spain theorem [cf. \textit{W. Rehder}, Int. J. Math. Math. Sci. 5, 813--816 (1982; Zbl 0503.47018); \textit{E. Albrecht} and \textit{P. G. Spain}, Proc. Am. Math. Soc. 128, No.~8, 2509--2511 (2000; Zbl 0951.47022)] says that, if a product of two bounded self-adjoint operators \(H, K\) is normal, then it is self-adjoint provided that the spectrum of \(K\) satisfies \(\sigma(K)\cap\sigma(-K)\subseteq\{0\}\). The author in [Proc. Am. Math. Soc. 131, No. 10, 3135--3141 (2003; Zbl 1049.47019)] extended that result to the case of unbounded operators. The paper under review is a continuation of the latter research. The author examines in detail a counterexample showing that merely assuming that \(HK\) has normal closure does not imply that \(HK\) has a self-adjoint closure (here, \(H\) and \(K\) are self-adjoint unbounded operators and \(K\) is positive). At the end, he gives an example of two unbounded self-adjoint operators \(H\) and \(K\) such that \(\sigma(K)\cap\sigma(-K)\subseteq\{0\}\), \(KH\) is normal but not self-adjoint. Thus the order of the operators \(H\) and \(K\) cannot be interchanged.