Some properties of the KMS-function (Q2708586)

Let, for \(j=0,1\), \(H_j=H^*_j \geq-w_j>-\infty\), be self-adjoint operators in the Hilbert spaces \({\mathcal H}_j\). Let \(T:D(T) \subset {\mathcal H}_1\to R(T) \subset{\mathcal H}_0\) be linear, \(v_j(t): =\exp(-tH_j)\), \(t\geq 0\). As some analog for the KMS-condition in the theory of von Neumann algebras, the author proves some so-called KMS-formulas for the representation of the operator \({\mathcal D}(t_0)T: =\int^{t_0}_0 V_0(u)Tv_1 (t_0-u)du\) in terms of the operators \(\Delta_1 :=(aI+H_0) V_0(t_0)T(aI+H_1)^{-1}\) and \(\Delta_2:=(aI+H_0)^{-1}T(aI+ H_1) V_1(t_0)\). As corollaries, the author establishes that if \(\Delta_1\) and \(\Delta_2\) are compact (resp., Hilbert-Schmidt, trace class), then so is the operator \({\mathcal D}(t_0)T\). Further, some convergence and approximation results for \({\mathcal D}(t_0)T\) are presented in terms of the operator \(t_0V_0(t_0/2) TV_1(t_0/2)\).NEWLINENEWLINEFor the entire collection see [Zbl 0957.00037].