Operator semigroup structure for conformable derivatives

Develop a well-defined operator semigroup structure for conformable derivative operators that scale the ordinary derivative by a temperature-dependent factor (e.g., T^(1−μ)), specifying generators, domains, and semigroup properties to provide a rigorous functional-analytic foundation for the conformable framework.

Background

The paper introduces a conformable derivative framework to model critical phenomena and applies it to modified Ginzburg–Landau equations and thermodynamic observables. While phenomenological results and data fits are presented, the authors highlight gaps in rigorous mathematical underpinnings of the conformable approach.

Among these, the need for an operator semigroup structure is central to establishing time or parameter evolution governed by conformable operators in a mathematically consistent way, ensuring existence, uniqueness, and continuity properties analogous to classical semigroups.