Non- (quantum) differentiable $C^1$-functions in the spaces with trivial Boyd indices

TL;DR

This paper constructs a specific $C^1$-function and operator in certain symmetric sequence spaces with trivial Boyd indices, demonstrating a failure of differentiability in the context of operator derivations.

Contribution

It introduces a novel example of a $C^1$-function in operator ideals where the functional calculus does not preserve domain inclusion under derivations.

Findings

Existence of a $C^1$-function with non-preservation of domain

Construction of an operator with specific derivation properties

Counterexample in symmetric sequence spaces with trivial Boyd indices

Abstract

If E is a separable symmetric sequence space with trivial Boyd indices and $\cC^E$ is the corresponding ideal of compact operators, then there exists a $C^1$-function $f_E$, a self-adjoint element $W\in \cC^E$ and a densely defined closed symmetric derivation $\delta$ on $\cC^E$ such that $W \in Dom \delta$, but $f_E(W) \notin Dom \delta$.