1702 CHAPTER 54. FUNCTIONAL ANALYSIS APPLICATIONS

n×n matrix as a special case. The identity Φ(t)Φ(s) = Φ(t + s) holds for any t,s ∈R andso is called a group of transformations. However, in the more general case, the identity onlyholds for t,s≥ 0 which is why it is called a semigroup. In this more general setting, I willcall it S (t). I am mostly following the presentation in Henry [63] in this short introduction.In what follows H will be a Banach space unless specified to be a Hilbert space. This newmaterial differs in letting A be only a closed densely defined operator. It might not be abounded operator.

These semigroups are useful in considering various partial differential equations whichcan be considered just like they were ordinary differential equations in the form u′+Au =f (u). The semigroups discussed here, when applied to actual examples, have the propertyof allowing one to begin with a very un-smooth initial condition, something in H, andmaking S (t)x in D(A) for all t > 0. When applied to partial differential equations, thistypically has the effect of making a solution t → S (t)x smoother for positive t than theinitial condition.

One can show that λ → (λ I−A)−1 is analytic on its so called resolvent set. Thisfollows from two things, the resolvant identity

(λ I−A)−1 (µI−A)−1 = (µ−λ )−1((λ I−A)−1− (µI−A)−1

)which follows from an observation that (µI−A), (λ I−A) are onto so the identity holds ifand only if

(λ I−A)−1 (µI−A)−1 (µI−A) = (µ−λ )−1((λ I−A)−1− (µI−A)−1

)(µI−A)

if and only if

(λ I−A)−1 = (µ−λ )−1((λ I−A)−1 (µI−A)− I

)= (µ−λ )−1

((λ I−A)−1 ((µ−λ ) I +(λ I−A))− I

)if and only if

(µ−λ )(λ I−A)−1 = (λ I−A)−1 ((µ−λ ) I +(λ I−A))− I

= (µ−λ )(λ I−A)−1 + I− I,

and an assumption that supλ

∥∥∥(λ I−A)−1 x∥∥∥ < ∞ for all λ near µ which by the Uniform

boundedness theorem implies∥∥∥(λ I−A)−1

∥∥∥ is bounded for λ near µ .

Thus I will always assume this resolvent λ → (λ I−A)−1 is analytic for λ on its resol-vent set, where this function is analytic. As to the resolvent set, the following describes itin this case of sectorial operators.

Definition 54.3.1 Let φ < π/2 and for a ∈ R, let Saφ denote the sector in the complexplane

{z ∈ C\{a} : |arg(z−a)| ≤ π−φ}