HILLE - YOSIDA Cantate
Theorem, Proof,
and
Applications
CHORAL [Theorem]
(1) Let A be closed and linear
with dense domain in X.
It generates a semigroup and we
can all relax.
It does so if it fulfills - that is
known to everybody
the condition of Hille and Yosida.
RECITATIV [Proof]
For the proof of the just heard
Theorem of Hille and of Yosida
we first remark the following:
The existence of the semigroup
is shown by bounded
approximation of A.
Thus, a sequence of uniformly
continuous semigroups is
obtained, which converges to the
semigroup T.
It remains to show, that A is the
generator of this semigroup.
This, however, is left to the
reader as an exercise.
CHORAL [Quod erat
demonstrandum]
ARIE [Applications]
Oh, what a lovely theorem! It can
be used every day.
By this theorem, generators are
identified by properties of their
resolvents.
Many bad PDE´s are solved by
that theorem.
In physics and also in space
travel there is no success
without Hille- Yosida.
Therefore, the Theorem is
beautiful and true.
FINALE [Applications II]
1. Let us now generate without
hesitation; approximation,
and rescaling is now trivial.
2. Now, everybody can solve the
problems, good or evil,
approximation, and rescaling
is now trivial.
3. Long live Hille and Yosida,
since they showed us this
theorem; approximation, and
rescaling is now trivial.