On weakly D-differentiable operators

TL;DR

This paper introduces the concept of weak D-differentiability for bounded operators relative to an unbounded self-adjoint operator D, providing equivalent conditions for this property.

Contribution

It formalizes the notion of weak D-differentiability and establishes multiple equivalent criteria for identifying such operators.

Findings

Characterization of weak D-differentiability through equivalent conditions

Extension of differentiability concepts to unbounded operators

Framework for analyzing operator dynamics in Hilbert spaces

Abstract

For an unbounded self-adjoint operator D on a Hilbert space H and a bounded operator a on H we say that a is weakly D-differentiable if for any pair of vectors x, y in H the function <exp(itD) a exp(-itD)x, y> is differentiable at t =0. We find several conditions which are all equivalent to weak D-differentiability.