Higher Weak Derivatives and Reflexive Algebras of Operators

TL;DR

This paper investigates higher weak derivatives of operators relative to a self-adjoint operator and characterizes the structure of associated reflexive operator algebras within von Neumann algebras.

Contribution

It introduces new characterizations of n-times weak D-differentiability and demonstrates that these operators form a reflexive algebra on an enlarged Hilbert space.

Findings

Characterizations of n-times weak D-differentiability.

Representation of weakly D-differentiable operators as reflexive algebras.

Application to von Neumann algebras and operator theory.

Abstract

Let D be a self-adjoint operator on a Hilbert space H and x a bounded operator on H. We say that x is n-times weakly D-differentiable, if for any pair of vectors a, b from H the function < exp(itD)x exp(-itD) a, b> is n-times differentiable. We give several characterizations of this property, among which one is original. The results are used to show, that for a von Neumann algebra M on H, the sub-algebra of n-times weakly D-differentiable operators has a representation as a reflexive algebra of operators on a bigger Hilbert space.