Consider a $C_0$-Semigroup $(T(t))_{t \ge 0}$ on the Banachspace $X$. Are there any conditions on $x \in X$ or the semigroup such that for any $t > 0$ it holds $T(t)x \in D(A)$? Here $(A,D(A))$ denotes the Generator with its domain $D(A)$.
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Let me ask a silly question. Consider the heat semigroup $e^{t\Delta}$ on $L^2(\mathbb R)$. Does it satisfy this property? (If not, there's little hope in general, IMHO) – Giuseppe Negro Oct 07 '17 at 12:14
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Indeed, it was silly, of course $e^{t\Delta}$ does satisfy the property. I made a bit of bibliographical research, see answer below. – Giuseppe Negro Oct 07 '17 at 13:32
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Tahnk you very much for your effort! – Oct 08 '17 at 09:15
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The semigroups which satisfy the property you mention are called immediately differentiable (see Engel-Nagel's book, Section 4.b, "Differentiable semigroups"). Corollary 4.15 provides a characterization in terms of a spectral property of the generator.
Analytic semigroups, which are holomorphic in the time variable on a sector in the complex plane, are immediately differentiable semigroups (see also this comment ). See Section 4.a of the aforementioned book.
Giuseppe Negro
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