What is an integral of operators?

Question

I am reading the book Semigroups of Linear Operatos and Applications to Partial Differential Equations which studies a uniformly continuous semigroup, this is a family $(T_t)_{t \geq 0}$ of bounded operators $T :X \to X$ on a Banachspace $X$ with

• $T(0)=Id$
• $T_{s+t}=T_s T_t$
• $ \lim_{t \to 0^+} || T_t - I || = 0$

They define the infinitesimal generator as $$ Ax :=\lim_{t \to 0^+} \frac{T_tx -x}{t} $$ for $x \in \mathcal{D}(A) := \{ x \in X \mid \lim_{t \to 0^+} \frac{T_tx -x}{t} \text{ exists} \}.$

Theorem 1.2 states

A linear operator A is the infinitesimal generator of a uniformly continuous semigroup if and only if A is a bounded linear operator

I am having difficulties to understand one direction of the proof, namely that for a uniformly continuous semigroup you find a bounded linear operator. The start of the proof goes like this:

Let $(T_t)_{t \geq 0}$ be a uniformly continuous semigroup of bounded linear operators on $X$. Fix $\rho >0 $ small enough, such that $$||I - \rho^{-1} \int_0^{\rho}T(s) \, ds|| < 1$$

But how is the integral of operators defined and why can you assure such a $\rho> 0$?

Answer 1

How is the integral of operators defined?

See Appendix E.5 in Evans book, where he extends

the notions of measurability, integrability, etc. to mappings $\mathbf{f} : [0, T]\to X $ where $T > 0$ and $X$ is a real Banach space, with norm $\|\;\|$.

See Appendix C in Engel's book where he gives

a brief introduction to the so-called Bochner integration of vector-valued functions.

See Chapter 1 in Vrabie's book where he defines and studies

the class of vector-valued measurable functions as well as the integral of such functions.

Why can you assure such a $\rho>0$?

Because $\displaystyle \lim_{\rho\to0^+}\left\|\frac{1}{\rho}\int_0^\rho T(s)\ ds-I\right\|=0$. See the proof here.