A generalization of Frenkel's formula

TL;DR

The paper develops an operator-level generalization of Frenkel's integral formula for divergences by introducing the operator divergence $\Delta(A\|B)=A(\log A-\log B)-B D\log[B](A)+B$ and proving two equivalent integral representations: $\Delta(A\|B)=\int_{-\infty}^{\infty} \frac{dt}{|t|(t-1)^2} ((1-t)A+tB)_-$ and $\Delta(A\|B)=\int_{1}^{\infty} (\gamma^{-1} O_{\gamma}(A\|B)+\gamma^{-2}O_{\gamma}(B\|A))\,d\gamma$ for $A\ge 0$, $B>0$. The authors provide a rigorous finite-dimensional proof and extend to infinite dimensions, introducing a convergence criterion $e<\infty$ (and $e_p<\infty$ for $p$-Schatten operators) to ensure well-defined operator limits via strong convergence of projected finite-rank truncations, with a dichotomy: finiteness when $A\le \tau B$ for some $\tau$, and divergence otherwise. The results generalize Frenkel's trace formula to operator-valued settings, offering operator-based integral tools that have potential applications in quantum information theory and operator analysis. Overall, the work broadens the scope of integral representations for divergences from scalars (traces) to operators, enabling new analytic handles in noncommutative settings.

Abstract

We generalize Frenkel's integral formula for traces of operators to operators. The resulting formula holds for bounded self-adjoint positive operators and $p$-Schatten class of compact positive operators.

Theorems & Definitions (16)

• Lemma 1
• proof
• Theorem 1
• proof
• Theorem 2
• Theorem 3
• proof
• Definition 1
• Lemma 2
• proof