Consider a semigroup $(T(t))_{t\in\mathbb{R}^+}$ generated by a densely defined strictly positive symmetric linear operator $A: D(A) \subset X \to X$, where $X$ is a Banach space with norm $\|\cdot\|$.

Besides, we introduce a Sobolev scale $(X_n)_{n\in\mathbb {Z}}$ induced by completion of $D(A^\infty)$ with respect to $\|\cdot\|_n:=\|A^n\cdot\|$, in particular $\|\cdot\|_0 = \|\cdot\|$.

My question is, under what conditions, there exists estimates such that

for $t\in (0,T)$, some $a >0$, $\|T(t)x\|_0 \leq C \|x\|_{-a}$, or $c \|x\|_{-a}\leq\|T(t)x\|_0 \leq C \|x\|_{-a}$.

In alternative, one may consider the weaker version, $t\in (t_0,T)$ with fixed $t_0>0$.