Are analytic semigroups $T(t)$ defined on a Banach space $X$, that is uniformly bounded with $0 \in \rho (A)$ the resolvent set of the infinitesimal generator A, have this property: $$ \lim_{t \to 0 } \|T(t) - I\|=0. \qquad (*)$$ I know that the property (*) is the definition of uniformly continuous semigroup, and in this case, $A$ bounded, which is not the case in my situation. In many papers concerning functional differential equations, the authors use this property while saying "by the strong continuity of $T(t)$ "? does anyone have an idea about this?
2026-03-26 14:43:18.1774536198
Analytic semigroup whose are uniformly bounded with 0 in resolvent of A
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