On the Spectral Theory of Regularized Quasi-semigroups

We investigate a time-dependent abstract Cauchy problem formulated as follows:
\begin{equation}
\begin{aligned}
x'(s) = A(s+t)x(s), \quad t, s \geq 0, \quad x(0) = Cx_0
\end{aligned}
\end{equation}

In this formulation, the function $x(s)$ represents an unknown function defined on the interval $[0,T]$ with values in a Banach space $X$. The operator $C$ is a bounded injective linear operator on $X$, while $A(s)$ denotes a closed linear operator in $X$ with constant domain $\mathcal{D}(A(t))=\mathcal{D}$ for all $t \geq 0$.

The solution to Equation (1) can be formally expressed as $x(t) = U(t,s)x_0$, where $\left\lbrace U(t,s)\right\rbrace_{t,s\geq 0}$ constitutes a two-parameter family of bounded linear operators on $X$, known as a $C$-quasi-semigroup or regularized quasi-semigroup. This concept, introduced by M. Janfada, generalizes classical $C_0$-semigroups to accommodate broader classes of evolution equations.

Our approach extends established spectral techniques from C₀-quasi-semigroups to the general framework of C-quasi-semigroups. The analysis investigates spectral properties of these operator families and their infinitesimal generators, which govern solution behavior.

We establish important spectral inclusion relationships between various spectra of C-quasi-semigroups and their infinitesimal generators. Specifically, we demonstrate spectral inclusions for Saphar, essentially Saphar, quasi-Fredholm, Kato, and essentially Kato spectra, providing comprehensive spectral characterization of these operators.

This work significantly extends the spectral theory of C₀-quasi-semigroups to regularized C-quasi-semigroups. The established spectral inclusion relations offer essential theoretical tools for analyzing regularized operators and their asymptotic properties, opening new perspectives for studying time-dependent differential equations in Banach spaces with regularization techniques.