Toeplitz properties of $\omega$-order preserving partial contraction mapping

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In this paper, spectral mapping theorem for the point spectrum on infinitesimal generator of a $C_0$-semigroup was further investigated. Toeplitz properties of semigroup considering $\omega$-order preserving partial contraction mapping ($\omega-OCP_n$) as a semigroup of linear operator was established to obtained new results. We also consider $A\in \omega-OCP_n$ which is the infinitesimal generator of a $C_{0}$-semigroup using the Spectral Mapping Theorem (SMT) to obtain the relationships between the spectrum of $A$ and the spectrum of each of the operators $\{T(t),~t\ge 0\}$.