Trace Formula For Two Variables

A natural generalization of Krein's theorem to a pair of commuting tuples $\left(H_1^0,H_2^0\right)$ and $\left(H_1,H_2\right)$ of bounded self-adjoint operators in a separable Hilbert space $\mathcal{H}$ with $H_j-H_j^0 = V_j\in \mathcal{B}_2(\mathcal{H})$(set of all Hilbert-Schmidt operators on $\mathcal{H}$) 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.