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Foundations of Double Operator Integrals with a Variant Approach to the Nonseparable Case

Robert Ferydouni, Daniel D. Spiegel

TL;DR

This work develops a self-contained foundation for double operator integrals (DOIs) on potentially nonseparable Hilbert spaces, including a streamlined construction of a projection-valued measure $\mathcal{G}$ and a robust, nonseparable integral projective tensor product framework with a complete norm. It proves a Daletskii–Krein formula for strongly differentiable perturbations via explicit Wiener-space kernels $\phi_t$, and establishes an extension of the DOI to trace-class and bounded operators with precise trace formulas and continuity properties. A key practical contribution is the application to quantum statistical mechanics, where a Hastings generator $D(s)$ yields an explicit evolution $P'(s)=i[D(s),P(s)]$ for spectral projections, with a unitary flow $U(s)$ implementing $P(s)=U(s)P(s_0)U(s)^*$. Overall, the paper provides new foundational tools—a new PVM construction, a nonseparable integral tensor product, and a Wiener-space–based DK formula—that enhance the applicability of operator-integral methods to perturbation theory and quantum many-body dynamics.

Abstract

We aim to give a self-contained and detailed yet simplified account of the foundations of the theory of double operator integrals, in order to provide an accessible entry point to the theory. We make two new contributions to these foundations: (1) a new proof of the existence of the product of two projection-valued measures, which allows for the definition of the double operator integral for Hilbert-Schmidt operators, and (2) a variant approach to the integral projective tensor product on arbitrary (not necessarily separable) Hilbert spaces using a somewhat more explicit norm than has previously been given. We prove the Daletskii-Krein formula for strongly differentiable perturbations of a densely-defined self-adjoint operator and conclude by reviewing an application of the theory to quantum statistical mechanics.

Foundations of Double Operator Integrals with a Variant Approach to the Nonseparable Case

TL;DR

This work develops a self-contained foundation for double operator integrals (DOIs) on potentially nonseparable Hilbert spaces, including a streamlined construction of a projection-valued measure and a robust, nonseparable integral projective tensor product framework with a complete norm. It proves a Daletskii–Krein formula for strongly differentiable perturbations via explicit Wiener-space kernels , and establishes an extension of the DOI to trace-class and bounded operators with precise trace formulas and continuity properties. A key practical contribution is the application to quantum statistical mechanics, where a Hastings generator yields an explicit evolution for spectral projections, with a unitary flow implementing . Overall, the paper provides new foundational tools—a new PVM construction, a nonseparable integral tensor product, and a Wiener-space–based DK formula—that enhance the applicability of operator-integral methods to perturbation theory and quantum many-body dynamics.

Abstract

We aim to give a self-contained and detailed yet simplified account of the foundations of the theory of double operator integrals, in order to provide an accessible entry point to the theory. We make two new contributions to these foundations: (1) a new proof of the existence of the product of two projection-valued measures, which allows for the definition of the double operator integral for Hilbert-Schmidt operators, and (2) a variant approach to the integral projective tensor product on arbitrary (not necessarily separable) Hilbert spaces using a somewhat more explicit norm than has previously been given. We prove the Daletskii-Krein formula for strongly differentiable perturbations of a densely-defined self-adjoint operator and conclude by reviewing an application of the theory to quantum statistical mechanics.

Paper Structure

This paper contains 6 sections, 35 theorems, 280 equations.

Table of Contents

  1. Introduction
  2. Constructing a PVM on Hilbert-Schmidt Operators
  3. Integral Projective Tensor Products
  4. The Double Operator Integral on TC and B
  5. The Daletskii-Krein Formula
  6. An Application from Quantum Statistical Mechanics

Key Result

Theorem 1

The set $\mathfrak{M}_0$ is a unital $*$-subalgebra of the $C^*$-algebra $B(X \times Y)$ of bounded measurable functions $X \times Y \to \mathbb{C}$. The function $\norm{\cdot}_{\mathfrak{M}_0}$ is a submultiplicative seminorm on $\mathfrak{M}_0$ invariant under complex conjugation. Defining $\pi:B( for every $\phi \in \mathfrak{M}_0$. Defining $\mathfrak{M} = \pi(\mathfrak{M}_0)$, the seminorm $\

Theorems & Definitions (79)

  • Theorem : \ref{['thm:integral_projective_tensor_product']}
  • Proposition 2.1: Birman_Solomyak_DoubleOperatorIntegral
  • proof
  • Proposition 2.2: Birman_Solomyak_DoubleOperatorIntegral
  • proof
  • Lemma 2.3: Birman_Solomyak_tensor_product_PVM
  • proof
  • Theorem 2.4
  • proof
  • Theorem 2.5: cf. Birman_Solomjak_SpectralTheory
  • ...and 69 more