With the theory of functions of a complex variable, it is time to consider the notion of analytic semigroups. These are better than continuous semigroups. I am mostly following the presentation in Henry [?]. In what follows H will be a Banach space unless specified to be a Hilbert space.
This sector is as shown below.
where εC < 1. Then A + B is also sectorial.
is bounded for λ ∈ Sbϕ. This proves the proposition.
It is an interesting proposition because when you have compact embeddings, such inequalities tend to hold.
Definition 53.2.3 Let ε > 0 and for a sectorial operator as defined above, let the contour γε,ϕ be as shown next where the orientation is also as shown by the arrow.
The little circle has radius ε in the above contour.
0 and S0ϕ. Note how the dotted line is at right angles to the solid line.
≡ I. It isn't necessary that ε be small, just that γε,ϕ not contain t. This is because the integrand in 53.2.6 is analytic.
Then it is necessary to show the above definition is well defined.
which is independent of t.
is bounded. This proves the Lemma.
is bounded for all positive real t. The following is the main result.
Theorem 53.2.6 Let A be a sectorial operator as defined in Definition 53.2.1 for the sector S0ϕ.
x is continuous at t = 0 for all x ∈ H.
Proof: Consider the first claim. This follows right away from the formula.
−1 −1 −1( − 1 −1) (λI − A) (μI − A) = (μ − λ) (λI − A ) − (μI − A) .
Now the following diagram might help in drawing some interesting conclusions.