Let $(T(t))_{t\geq 0}$ be a $C_0$ semigroup of bounded linear operators on a Banach space $X$ and denote its generator by $A$. A fundamental problem to decide whether the Drazin spectrum of each operator $T(t)$ can be obtained from the Drazin spectrum of $A$. In particular, one hopes that the Drazin Spectral Mapping Theorem holds, i.e., $e^{t \sigma_{D}(A)}=\sigma_{D}(T(t))\backslash \{0\}$ for all $t \geq 0$.
In this work we show that the spectral inclusion of semigroups hold for descend, ascent, essential descend , essential ascent, Drazin, Kato, and essential Kato.
Let (C(t))
t∈ℝ be a strongly continuous cosine family and A be its infinitesimal generator. In this work, we prove that, if C(t) – cosh λt is semi-Fredholm (resp. semi-Browder, Drazin inversible, left essentially Drazin and right essentially Drazin invertible) operator and λt ∉ iπℤ, then A – λ
2 is also. We show by counterexample that the converse is false in general.
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