The Variant Spectrum Of C₀-semigroup And Spectral Properties Of Generation Operator

For a strongly continuous semigroup(T(t)t≥0 with the generator A,in general,the spectral mapping theorem σ(T(t))\{0} = etσ(A)may fail.We introduce its critical spectrum σcrit(T(t)),thus the spectral mapping theorem can be generalized to the formσ(T(t)\{0} =etσ(A)(?)σcrit(T(t))\{0},t≥0,(1)We called equation(1)as spectral mapping theorem of critical spectrum,the intersection of etσ(A)and σcrit(T(t))may be nonempty,hence for a C0 semigroup,its critical spectrum is nonempty cannot illustrate the spectral mapping theorem is not valid.We define its variant spectrum σv(T(t))the spectral mapping theorem can be further generalized to the following formσ(T(t))\{0}=etσ(A)(?)σv(T(t))\{0},t≥0,(2)We called equation(2)as spectral mapping theorem of variant spectrum,the intersection of etσ(A)and σv(T(t))must be empty,so for a C0 semigroup,its variant spectrum is nonempty if and only if spectral mapping theorem is not valid.Equation(1)and equation(2)are both right for all strongly continuous semigroups,they are collectively called generalized spectral mapping theorem.