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

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- Solution e

Solution

The family

U (t) = e^{- ω t} U (t)

inherits the identity, semigroup property, and strong continuity from

U

, while

∥ U (t) u ∥ \leq M ∥ u ∥.

Its difference quotient satisfies

\frac{U ( h ) u - u}{h} = e^{- ωh} \frac{U ( h ) u - u}{h} + \frac{e ^{- ωh} - 1}{h} u ⟶ A u - ω u

for

u \in D (A)

. Conversely, existence of this limit implies existence of the generator limit for

U

, so

D (A) = D (A)

and

A = A - ω 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;
$(0, \infty) \subseteq ρ (B)$ ;
for every $λ > 0$ and integer $n \geq 1$ ,

∥ λ^{n} (λ I - B)^{- n} ∥ \leq M .

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