Description of \(\operatorname {AlgLat}T/W(T)\) for certain extensions of selfadjoint operators (Q1595114)

Let \(H\) and \(K\) be Hilbert spaces. The operator \(T\) mentioned in the title is of the form \(T=\left(\begin{smallmatrix} A&X\\ 0&N\end{smallmatrix}\right)\), where \(A\) is a selfadjoint operator in \({\mathcal L}(H)\) such that \(0\) is not in the point spectrum of \(A\), \(N\in {\mathcal L}(K)\) is a cyclic nilpotent operator of order \(n\), and \(X\) belongs to \({\mathcal L}(K, H)\). The symbol \(W(T)\) in the title means the weak closure of the polynomials in \(T\). Let \(e\in K\) be a cyclic vector for \(N\). The description mentioned in the title is obtained under the condition that the minimal closed subspace invariand under \(A\) and containing \(T^n e\) is of finite codimension in \(H\). As a consequence it is shown that, under the above conditions, the quotient mentioned in the title is of dimension at most \(n(n-1)/2\), and a necessary and sufficient is given for this dimension to be \(0\).