The family
$$
\widehat U(t)=e^{-\omega t}U(t)
$$
inherits the identity, <semigroup property>, and from $U$, while
$$
\|\widehat U(t)u\|\leq M\|u\|.
$$
Its difference quotient satisfies
$$
\frac{\widehat U(h)u-u}{h}
=e^{-\omega h}\frac{U(h)u-u}{h}
+\frac{e^{-\omega h}-1}{h}u\longrightarrow Au-\omega u
$$
for $u\in D(A)$. Conversely, existence of this limit implies existence of the generator limit for $U$, so $D(\widehat A)=D(A)$ and
$$
\boxed{\widehat A=A-\omega I}.
$$
This is the <exponentially shifted semigroup> construction.

The <Hille-Yosida theorem> in the uniformly bounded case says that a <linear operator> $B$ on a <Banach space> generates a <C0-semigroup> with $\|e^{tB}\|\leq M$ if and only if:

* $B$ is a <closed linear operator> whose domain is a <dense subset> of the <Banach space>;
* $(0,\infty)\subseteq\rho(B)$;
* for every $\lambda>0$ and integer $n\geq1$,
$$
\boxed{\|\lambda^n(\lambda I-B)^{-n}\|\leq M}.
$$

Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 319 / 1 / e / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-319/1/e/solution