Trace Formula For Two Variables

A natural generalization of Krein’s theorem to a pair of commuting tuples $(H_1^0,H_2^0)$ and $(H_1,H_2)$ of bounded self-adjoint operators in a separable Hilbert space $\mathcal{H}$ with $H_j-H_j^0 = V_j\in $\mathcal{B}$_2($\mathcal{H}$)$(setofallHilbert-Schmidtoperatorson$$\mathcal{H}$$)for$j=1,2,$ leads to a Stokes-like formula under trace. A major ingredient in the proof is the finite-dimensional approximation result for commuting self-adjoint n-tuples of operators, a generalization of Weyl-von Neumann-Berg’s theorem.