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

TL;DR

This paper refines the theoretical foundations of double operator integrals, introducing new proofs and approaches that extend applicability to nonseparable Hilbert spaces and include practical applications in quantum statistical mechanics.

Contribution

It provides a new proof for the product of projection-valued measures and a variant approach to the integral projective tensor product for nonseparable Hilbert spaces.

Findings

Established the existence of the product of projection-valued measures.

Developed a new approach to the integral projective tensor product for nonseparable spaces.

Proved the Daletskii-Krein formula for strongly differentiable perturbations.

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.