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1.4. Hilbert space
The operator A − I maps D(A) onto X and so for all x ∈ X there exists a
y ∈ D(A) such that y = (A−I) −1 x. We can write relation (1.16) as follows.
〈Adx, Adx〉 − 〈x, x〉 ≤ 0, for all x ∈ X,
which implies inequality (1.14).
Lemma 1.8 implies that Ad is a contraction and this completes the proof.
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Another class of operators for which the relation between the boundedness
of the semigroup and the boundedness of the powers of the cogenerator is
known, is the class of analytic semigroups.
First we define for α ∈ (0, π
2 ) the sector ∆α in the complex plane by
∆α := {t ∈ C | | arg(t)| < α, t �= 0} .
A C0-semigroup (e At )t≥0 is analytic if it can be continued analytically into
∆α for an α > 0.
Lemma 1.10 Operator A generates the analytic semigroup e At and �e At � ≤
M for all t ∈ ∆α if and only if there exists an m ≥ 0 such that
�(sI − A) −1 � ≤ m
π
, for all complex s with | arg(s)| < + α.
|s| 2
For the proof we refer to Pazy, [30, Theorem 2.5.2].
Theorem 1.11 Let A generate an analytic C0-semigroup on the Hilbert
space X and let Ad be its cogenerator. If (e At )t≥0 is bounded, then the
powers of the cogenerator (A n d )n∈N are bounded as well.
For the proof we refer to [21, Theorem 6.1]. See also Chapter 7
There is a relation from the discrete-time system to the continuous-time
system as well. This relation involves bounded and strongly stable power
sequences. In Section 2.1 and Section 2.2 we give a definitions of these
properties.
Theorem 1.12 Let Ad be the Cayley transform of operator A and the powers
(An d )n∈N are bounded. If −1 is not an eigenvalue of Ad and A∗ d , the
adjoint of Ad, and if there exist a C > 0 and δ > 0 such that
|µ + 1|�R(µ, Ad)� ≤ C, for all |µ + 1| ≤ δ, Re(µ) < −1.
then A is generates a bounded semigroup, which is analytic.
Moreover, if (A n d )n∈N is strongly stable, so is (e At )t≥0.
For the proof we refer to [21, Theorem 7.1].
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