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 $t\_0>0$.



Estimate of semigroup with dual norm?
First, if $A$ is symmetric, then $X$ should be a Hilbert space, but I remain in a Banach space. 


If 
$$\|T(t)x\| \leq C\|A^{-1} x\|$$ 
holds for all $x\in X$, then using the substitution $y=A^{-1}x$, you obtain
$$\|AT(t)y\|\leq C\|y\|.$$


But by Theorem 5.3 in Section 2.5 of


*A. Pazy*, MR 710486 [**Semigroups of linear operators and applications to partial differential equations**](http://dx.doi.org/10.1007/978-1-4612-5561-1), ISBN: 0-387-90845-5.


this implies that $A$ is bounded. Hence, I do not see much of a chance for your inequalities to hold.


