Random Multiple Operator Integrals

TL;DR

This paper extends the theory of Double Operator Integrals to Multiple Operator Integrals, generalizing Schur multipliers and providing new tail bounds for norms of random operator derivatives and differences.

Contribution

It introduces a generalized framework for MOIs using Schur multipliers, extending previous work from finite-dimensional to infinite-dimensional random operators.

Findings

Generalized Schur multipliers to MOIs via closure spaces.

Expressed functions on compact sets as limits of tensor products.

Derived tail bounds for norms of random operator derivatives and remainders.

Abstract

The introduction of Schur multipliers into the context of Double Operator Integrals (DOIs) was proposed by V. V. Peller in 1985. This work extends theorem on Schur multipliers from measurable functions to their closure space and generalizes the definition of DOIs to Multiple Operator Integrals (MOIs) for integrand functions as Schur multipliersconstructible by taking the limit of projective tensor product and by taking the limit of integral projective tensor product. According to such closure space construction for integrand functions, we demonstrate that any function defined on a compact set of a Euclidean space can be expressed by taking the limit of the projective tensor product of linear functions. We also generalize previous works about random DOIs with respect to finite dimensional operators, tensors, to MOIs with respect to random operators, which are defined from spectral decomposition perspectives. Based on random MOIs definitions and their properties, we derive several tail bounds for norms of higher random operator derivatives, higher random operator difference and Taylor remainder of random operator-valued functions.