A Discrete Resolvent Framework for Operator Splitting in Delay Differential Equations

We establish a discrete operator--theoretic framework for the analysis of implicit Euler and Lie--Trotter splitting schemes for delay differential equations (DDEs). Both schemes are formulated in terms of discrete resolvent operators acting on product spaces that encode the present state together with the history variable. The analysis is carried out entirely at the level of discrete propagators and does not presuppose the existence of a $C_0$-semigroup or an evolution family generated by the underlying delay operator. Convergence of Lie--Trotter splitting toward implicit Euler on finite time intervals is shown to follow from two structural ingredients: local defect estimates on fractional interpolation spaces and suitable discrete stability properties of the associated operator products. The framework applies to both autonomous and non-autonomous delay equations. In the non-sectorial case, additional discrete stability assumptions (such as power-boundedness or Ritt-type conditions) are required to control the accumulation of local errors. In contrast, when the principal operator is sectorial and generates an analytic semigroup, analytic smoothing suppresses error growth and preserves the fractional convergence order on interpolation spaces. This discrete resolvent perspective provides a unified approach to operator splitting for delay equations, including regimes in which classical semigroup well-posedness theory is unavailable. This version presents a substantially revised and restructured analysis, including a unified discrete stability framework and extended numerical experiments.