On certain multivariable subnormal weighted shifts and their duals (Q2861262)

A commuting \(m\)-tuple \(S\) on \(\mathcal H\) is \textit{subnormal} if it admits a normal extension possibly on a larger Hilbert space \(\mathcal K.\) Note that \(N^*\) is a normal tuple with invariant subspace \(\mathcal K \ominus \mathcal H.\) The tuple \(\tilde{S}:=N^*|_{\mathcal K \ominus \mathcal H}\) is referred to as the \textit{dual} of \(S.\) This notion was introduced by \textit{J. Conway} in single variable [J. Oper. Theory 5, 195--211 (1981; Zbl 0469.47020)] and \textit{A. Athavale} in several variables [Integral Equations Oper. Theory 12, No. 3, 305--323 (1989; Zbl 0682.47011)]. A subnormal tuple is said to be \textit{self-dual} if \(S\) is unitary equivalent to \(\tilde{S}\). A basic result in the theory of duals of subnormal operators, due to J. Conway, characterizes self-dual weighted shifts \(S\) of norm \(1\): Either \(S\) is the unilateral shift \(U\) with respect to an orthonormal basis \(\{e_n\}_{n \geq 0}\) or \(S=U+K,\) where \(K\) is the rank one operator which sends \(e_0\) to \((1/\sqrt{2}-1)e_1\) and \(Ke_n=0\) for \(n \geq 1.\)NEWLINENEWLINEThe main result in the paper under review provides a family of subnormal \(m\)-shifts \(S\) \((m \geq 2)\) (which in particular includes Fredholm subnormal \(m\)-shifts) which are far from being self-dual: \(S\) is not even similar to its dual \(\tilde{S}\). The arguments in the proof are based on examining the dimensions of the cohomology vector spaces associated with the cochain Koszul complexes of \(S\) and its dual \(\tilde{S}\).