Consider two densely defined, strictly positive, self-adjoint operators $A$ and $B$ with the following property $$\|A^k x\| \simeq \|B^k x\|, \forall x \in D(A^k)= D(B^k),$$ for $k=1,2,\cdots, M$, where $M$ can be $\infty$.

Do we have for fixed $t>0$, $$\|T(t) x \| \simeq \|S(t) x\|, $$ where $T(t),S(t)$ are the semigroups generated by $A,B$?

Does it hold only when $M = \infty$? Or does it hold under some extra assumptions?

Notation: $a \simeq b$ means there exist a positive constant $C$ such that $\frac{1}{C}a \leq b \leq C a.$

Consider two densely defined, strictly positive, self-adjoint operators $A$ and $B$ with the following property $$\|A^k x\| \simeq \|B^k x\|, \quad\forall x \in D(A^k)= D(B^k),$$ for $k=1,2,\cdots, M$, where $M$ can be $\infty$.

Do we have for fixed $t>0$, $$\|T(t) x \| \simeq \|S(t) x\|, $$ where $T(t),S(t)$ are the semigroups generated by $A,B$?

Does it hold only when $M = \infty$? Or does it hold under some extra assumptions?

Notation: $a \simeq b$ means there exist a positive constant $C$ such that $\frac{1}{C}a \leq b \leq C a.$