Possible bound on generator in terms of the semigroup it generates?

Question

Suppose $T$ is a $C_0$-semigroup with $\omega \in \mathbb{R}$ and $M \geq 0$ such that $$\| T(t) \| \leq M e^{\omega t}.$$ Let $A$ be the generator for $T$ and let $\xi_1, ..., \xi_n >0$ be constants. I want to get some form of a bound of $\| (\xi_1 \cdots \xi_n)^{-1} (\xi_1^{-1} - A)^{-1} \cdots (\xi_n^{-1} - A)^{-1} \|$ in terms of $T$. I've attempted the following. $$\| (\xi_1 \cdots \xi_n)^{-1} (\xi_1^{-1} - A)^{-1} \cdots (\xi_n^{-1} - A)^{-1} \| \leq \frac{1}{\xi_1 \cdots \xi_n} \prod_{i=1}^n \| (\xi_i - A)^{-1} \| \\ \leq \frac{1}{\xi_1 \cdots \xi_n} \prod_{i=1}^n \frac{M}{\text{Re}(\xi_i^{-1}) - \omega}$$ Can I get this to be less than the supremum of $T$?

Answer 1

Not a complete answer, but too long to be posted as a comment.

1. You have an operator $A$ which is the infinitesimal generator of a $C_0$-semigroup $T=\{T(t)\}$ satisfying $$\|T(t)\|\leq Me^{\omega t}$$ for $\omega\in\mathbb R$ and $M\geq 0$.

2. You want to get a bound for $\left\|\prod \xi_i^{-1}(\xi_i^{-1}-A)^{-1}\right\|$, where $\xi_1,...,\xi_n$ are arbitrary positive constants.

Given only the informations in 1, I can't even see how to ensure that $(\xi_i^{-1}-A)$ is invertible for arbitrary $\xi_i>0$. So, I'm going to consider an extra hypothesis:

Assume that $\omega\leq 0$.

Note. This hypothesis is not totally restrictive. There are important cases in which it is fulfilled, for example, if $T$ is of contractions or exponentially stable (but I have no idea if it is relevant for you).

Under this assumption, the Feller-Miyadera-Phillips Theorem implies that $(0,\infty)\subset \rho (A)$ with $$\|(\lambda-A)^{-1}\|\leq \frac{M}{\lambda-\omega},\quad\forall \ \lambda>0$$

and thus, as in your calculation, we get the bound

$$\left\|\prod \xi_i^{-1}(\xi_i^{-1}-A)^{-1}\right\| \leq \prod \xi_i^{-1}\left\| (\xi_i^{-1}-A)^{-1}\right\| \leq \prod \xi_i^{-1}\frac{M}{\xi_i^{-1}-\omega} = \prod \frac{M}{1-\xi_i\omega}\leq M^n. $$

Remark. The main point of the above argument is that $\xi_i>0$ (as you want) and $\omega\leq 0$ (as I assumed) imply $\xi_i^{-1}>\omega$ and this makes possible to apply the Feller-Miyadera-Phillips Theorem. So, the above estimate is valid provided that $\xi_1^{-1},...,\xi_n^{-1}>\omega$ (which happens for arbitrary $\xi_1,...,\xi_n>0$ if $T$ is of contraction or exponentially stable, as I already said).