Unbounded weighted shifts and subnormality (Q1122114)

The operator \(S=(1/\sqrt{2})(x-d/dx)\) defined on the linear span of the Hermite functions is an example of an unbounded subnormal weighted shift [cf. the authors, J. Oper. Theory 14, 31--55 (1985; Zbl 0613.47022)]. It is shown that certain products of unbounded subnormal weighted shifts are again subnormal, for example the products \(R^{*pt}S^{pt}T\), \(TR^{pt}S^{*pt}\), \(TR^{*pt}S^{pt}\), \(R^{pt}S^{*pt}T\) for integers \(p\geq 0\), \(t\geq 1\) if \(R,S\) are usual weighted shifts and \(T\) is a weighted \(t\)-shift (i.e. it shifts \(t\)-times)