Suppose ${\lbrace T(t)\rbrace}_{t \ge 0} $ be a semigroup of operators on some Banach space $X$. How to prove semigroup $ T(t) $ is bounded dissipative ? Here bounded dissipative means, there exist a bounded set $B$ in $X$ which attract every bounded set of $X$ under $T(t) \quad \text{i.e} \quad \text{ dist}(T(t) \tilde{B}, B) \rightarrow 0$ as $ t \rightarrow \infty$ for every bounded set $\tilde{B}$ in $X$. I am following the book of Jack K. Hale for study of dissipative systems. I am not finding any example in this book for this purpose. For $\alpha >0$, suppose our semigroup is $\text{e}^{-\alpha t}T(t)$ , where $T(t)$ is the semigroup generated by Dirichlet laplacian in $L^{2}(0,1)$. For this how we will proceed.
2026-03-25 19:03:42.1774465422
How to prove semigroup of linear operators is bounded dissipative?
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