Determinant formula and trace formula for some finite rank perturbation of Heisenberg commutation relation (Q930463)

Following his work [Integral Equations Oper.\ Theory 37, No.\,4, 487--504 (2000; Zbl 0981.47021)], the author considers the class \(\mathcal{F(H)}\) of unbounded linear operators \(A\) in a separable Hilbert space \(\mathcal H\) such that \([A,A^*]x = x+Dx,\) for \(x\) in a certain dense linear manifold \(\mathcal {D = D}(A,D)\subset \mathcal H\) and a finite rank perturbation \(D\). Under some conditions, entailing equivalence of \(A\) to a (direct orthogonal sum of) multiplication operator(s) in the Bargmann--Wigner type weighted space of entire functions on \(\mathbb C\), one has \(\text{Det}(e^{sA^*} e^{tA}e^{-sA^*}e^{-t(A+s)}) = \exp(st\text{Tr}(D))\) for \(s,t \in\mathbb C\). Moreover, in the spirit of the author's previous paper [Integral Equations Oper.\ Theory 56, No.\,4, 571--585 (2006; Zbl 1109.47013)], he shows that \(\text{Tr}(e^{sA^*}Q(A)e^{-sA^*} - Q(A+s)) = \text{Tr}((Q(A+s) - Q(A))D)\) for a polynomial \(Q\).