\(C_o\) -semigroups
In this chapter we concentrate on strongly continuous or more specifically \(C_o\) -semigroups of bounded operators on a Banach space. The notion of the generator of a \(C_o\) -semigroup is introduced and their properties are dealt with in detail.
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C0-semigroups In this chapter we concentrate on strongly continuous or more specifically C0 semigroups of bounded operators on a Banach space. The notion of the generator of a C0 -semigroup is introduced and their properties are dealt with in detail.
2.1 Introduction Consider a function T : [0, ∞) → Mn (C), satisfying the following properties: (i) T (0) = I, (ii) T (t + s) = T (t)T (s) ∀ t, s ≥ 0 and (iii) T (t) → I as t → 0. Then it is not difficult to see that the map t → T (t) is differentiable and T (t) = eAt for some A ∈ Mn (C) (Exercise 2.1.1). Here continuity, together with the semigroup property implies differentiability. This implication carries over to the infinite-dimensional case also, but with a qualification – A may not be defined everywhere. (Recall that A ∈ Mn (C) may be considered as a linear operator on Cn , defined everywhere). We prove the above assertion in this section. The following simple result will be used repeatedly in the text and is given for the sake of completeness. Lemma 2.1.2. Let X be a Banach space and let f ∈ [0, a] → X be a continuous function. Then lim+ t
t→0
−1
t
f (s) ds = f (0). 0
© Springer Nature Singapore Pte Ltd. 2017 and Hindustan Book Agency 2017 K.B. Sinha and S. Srivastava, Theory of Semigroups and Applications, Texts and Readings in Mathematics, DOI 10.1007/978-981-10-4864-7_2
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C0 -semigroups
Proof. Note that the integral exists as a Riemann integral as well as a Bochner integral (Lemma 1.2.5) and that t
−1
t
0
t −1 f (s) ds − f (0) = t [f (s) − f (0)] ds. 0
Since continuity implies uniform continuity on a compact interval, given > 0, one can find a δ > 0 such that f (s)−f (0) < whenever 0 < s < δ. Therefore, −1 t
0
t
f (s) ds − f (0) ≤ t−1
t
0
f (s) − f (0) ds <
for 0 < t < δ.
2.2 The generator Assume that T : (0, ∞) → B(X) is a semigroup of operators and is strongly continuous on (0, ∞). The infinitesimal generator A0 of T is defined in the following manner. Set Aη x
=
A0 x
=
T (η)x − x , η > 0, and η lim+ Aη x,
(2.1) (2.2)
η→0
whenever the limit exists. From now on we shall refer to the infinitesimal generator as simply the generator. The domain D(A0 ) of A0 is the set of all x ∈ X such that the limit in (2.2) above exists. Then D(A0 ) is a linear subspace and A0 is a linear operator. In general, the operator A0 may not be closed, nor densely defined. But, D(A0 ) is always non-empty: Lemma 2.2.1. D(A0 ) is non-empty.
Proof. For y ∈ X and 0 < α < β < ∞, set xα,β = Lemma 1.2.5. We shall establish that
β
α limη→0 Aη xα,β
T (t)y dt, which exists by exists, thus proving the
result. Indeed for η > 0, by a change of variable in one of the integrals, we have
2.2. The generator
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that Aη xα,β = =
1 η 1 η
1 = η
β
T (t + η)y dt −
α
β+η β
η 0
T (t)y dt −
1 η
1 η
β
T (t)y dt α α+η
T (t)y dt α
1 T (t + β)y dt − η
η
T (t + α)y dt 0
−→ (T (β)y − T (α)y), as η → 0+ , by Lemma 2.1.2.
Next, we set, for α > 0, Xα = T (α)(X) and X0 =
Xα .
α>0
The semigroup property clearly
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