Spectral analysis of main operator of Rotenberg equation with time delay

In 1983, Rotenberg firstly proposed the Rotenberg model, which has attracted wide attention. For Rotenberg model, many previous studies have been carried out, but the time delay of cell division was not considered in previous studies. In order to describe cell proliferation better and more accurately, on the basis of previous studies, we consider the time delay of cell division, make some modifications to the original Rotenberg model, and prove that transport operator A_H can generate a C₀ semigroup. The spectrum of the corresponding transport operator is analyzed, and it is only consists of finite discrete eigenvalues with finite algebraic multiplicity in the region $\mathrm{\Gamma }=\sigma \left({\mathrm{A}}_{\mathrm{H}}\right)\cap \{\lambda \in C\mid \mathrm{Re}\lambda >\gamma \}\left(\gamma >{\lambda }_0\right)$.