Prelim Semigroup Notes Harry Gaebler 1 C0-semigroups Unless otherwise stated, we will assume throughout that X is a complex Banach space, with L (X) denoting the Banach space of continuous linear operators on X. Definition. (Semigroups) By a semigroup, we mean a one-parameter family of continuous linear operators, fT (t)gt≥0 ⊂ L (X), such that, 1. T (0) = I, where I is the identity map on X. 2. For all s; t ≥ 0, we have T (s + t) = T (s) ◦ T (t). Additionally, if one has limt!0+ T (t)x = x for every x 2 X, then the semigroup is said to be \strongly continuous," or C0, for short. To every C0 semigroup, we associate another linear map called its generator, which describes how the semigroup is affecting initial data in X. Definition. (Infinitesimal Generator) Given a C0-semigroup, fT (t)gt≥0, its in- finitesimal generator is the map A : D(A) ! X, where n T (t)x − x o D(A) = x 2 X lim exists t!0+ t and Ax is defined to be the value of this found limit for x 2 D(A). It is quite clear that D(A) is a subspace of X, and further, that A is a (not necessarily continuous) linear map. C0-semigroups and their generators enjoy some rather pleasant basic properties, which we will now elucidate. Lemma. (Continuity of Orbits) Given a C0-semigroup, fT (t)gt≥0, we have that the orbit map t 7! T (t)x is continuous for each x 2 X. Proof. Apply the semigroup property and then strong continuity. Lemma. (Exponential Bounds) Given a C0-semigroup, fT (t)gt≥0, there exists constants M ≥ 1 and ! 2 R such that for every t ≥ 0, jjT (t)jj ≤ Me!t. 1 Proof. Suppose for a contradiction that for every n 2 N, there exists a point 1 tn 2 [0; n ] such that jjT (tn)jj ≥ n. Now, defining fT (tn) j n 2 Ng ⊂ L (X) we see that for each x 2 X, sup jjT (t )xjj ≤ sup jjT (t)xjj < 1 by the n2N n [0;1] continuity of the orbit map. Then, by uniform boundedness, it follows that sup jjT (t )jj < 1, but this is an obvious contradiction. n2N n Specifically, we are now afforded an n0 2 N such that jjT (t)jj < n0 for every 1 1 t 2 [0; ]. Then, fixing t∗ ≥ 0 arbitrarily, we may write t∗ = k + r for some n0 n0 1 k 2 0 and r 2 [0; ), which yields that, N n0 1 k+1 n h 1 io jjT (t∗)jj = T k + r ≤ M M = sup jjT (t)jj t 2 0; n0 n0 k ln(M ) !t∗ = Me ≤ Me ! = n0 ln(M) and since M ≥ 1 and ! 2 R do not depend on t∗, we are done. The next lemma rounds out our basic \toolbox" of techniques for manipu- lating C0-semigroups and their generators. Lemma. (Toolbox) Given a C0-semigroup, fT (t)gt≥0, and its generator A : D(A) ! X, we may conclude the following: 1. For every x 2 X and t ≥ 0, we have 1 R t+h + (a) h t T (s)xds ! T (t)x as h ! 0 R t R t (b) 0 T (s)xds 2 D(A) and A( 0 T (s)xds) = T (t)x − x 2. For every x 2 D(A) and t ≥ 0, we have d (a) T (t)x 2 D(A) and dt T (t)x = AT (t)x = T (t)Ax R t R t (b) For every 0 ≤ s ≤ t, we have s T (τ)Axdτ = s AT (τ)xdτ = T (t)x− T (s)x 3. A is densely-defined and closed. Proof. Continuity of the orbit map implies 1 (a), and 1 (b) is a straightforward application of 1 (a). Next, 2 (a) follows from the semigroup property, and 2 (b) is simply the fundamental theorem of calculus. Lastly, 3 requires just a little bit of unpacking. First, fix x 2 X, and define, Z n−1 xn = n T (t)xdt (n 2 N) 0 so that xn 2 D(A) by 1 (b), and xn ! x by 1 (a) as n ! 1, meaning that A is densely-defined. Then, supposing xn ! y 2 X and Axn ! z 2 it follows by 2 (b) that for h > 0, we have Z h T (h)xn − xn 1 = T (t)Axndt h h 0 !h and since jjT (t)Axn − T (t)zjj ≤ Me jjAxn − zjj ! 0 as n ! 1, it follows by uniform convergence on [0; h] that, T (h)y − y 1 Z h = T (t)zdt h h 0 so taking h ! 0+, it follows by 1 (a) that y 2 D(A) and Ay = z, meaning that A is a closed operator, as required. The main idea behind semigroup theory is that it allows us to solve the so- called \Abstract Cauchy Problem" (ACP), which is the initial value problem, ( u0(t) = Au(t) t ≥ 0 u(0) = u0 u0 2 D(A) for some linear map A and unknown function u : [0; 1) ! X. We are always hoping to show that A generates a C0-semigroup, in which case the ACP is well-posed, and the solution is given via a \semigroup representation," u(t) = T (t)u0 (t ≥ 0) Recall from ODE theory that we should have T (t) ≈ etA, and it is easy to see tA that A 2 L (X) will generate fe gt≥0. Conversely, it is straightforward to show that given an exponential semigroup, its generator must be the exponentiated bounded operator. More generally, the collection of exponential semigroups con- stitute a subset of strongly continuous semigroups which are said to be uniformly continuous. Definition. (UC Semigroups) A strongly continuous semigroup, fT (t)gt≥0, is called uniformly continuous if limt!0+ jjT (t) − Ijj = 0. The following lemma is an important general fact, and we will use it here in the context of showing that the exponential semigroups and UC semigroups coincide. Lemma. (Von Neumann) If B1;B2 2 L (X) where B1 is an invertible map, −1 −1 and jjB2jj < jjB1 jj , then B1 − B2 2 L (X) is an invertible operator. Proof. We'll split this proof into two steps. The first step is to consider the case where B1 ≡ I and jjB2jj < 1. In this case, n n n X j X j X j+1 n+1 (I − B2) B2 = B2 − B2 = I − B2 (n 2 N) j=0 j=0 j=0 3 P1 j and taking n ! 1, it follows that (I −B2) j=0 B2 = I, and as this sum is also P1 j clearly a left inverse of I − B2, we have that I − B2 is invertible and j=0 B2 is its inverse. We next consider the general case, −1 B1 − B2 = B1(I − B1 B2) −1 and since jjB1 B2jj < 1, the LHS must be invertible, so B1 − B2 is invertible −1 −1 −1 −1 with inverse given by, (B1 − B2) = (I − B1 B2) B1 . Theorem. (EXP $ UC) The collection of exponential semigroups exactly coincides with the collection of UC semigroups, that is tA ffe gt≥0 j A 2 L (X)g = fUC semigroupsg Proof. The ⊆ containment is obvious, and for the ⊇ containment, suppose that fT (t)gt≥0 is UC and note that for a sufficiently small r > 0, we have 1 Z r 1 Z r I − T (t)dt ≤ jjI − T (t)jjdt < 1 r 0 r 0 R r so by the Von Neumann lemma, it follows that 0 T (t)dt must be invertible. Then, fixing x 2 X and h 2 (0; r), we have T (h)x − x h Z r i−1h Z r iT (h)x − x = T (t)dt T (t)dt h 0 0 h h Z r i−1h 1 Z r+h 1 Z h i = T (t)dt T (t)xdt − T (t)xdt 0 h r h 0 so letting G : D(G) ! X generate fT (t)gt≥0, one has X ⊆ D(G), and hence, R r −1 G = [ 0 T (t)dt] [T (r) − I] 2 L (X). Our results about UC semigroups now allow us to characterize the generators of arbitrary C0-semigroups, and this is hugely important because it tells us when the ACP is well-posed. Definition. (Resolvent and Spectrum) The resolvent set of a closed operator, A : D(A) ! X, is the set given by ρ(A) = fλ 2 C j λ − A : D(A) ! X is bijectiveg and its complement, σ(A) = C n ρ(A), is said to be the spectrum of A. Note that we do not assume the density of D(A) in X - only the closedness of A is needed for a reasonable spectral theory. For each λ 2 ρ(A), we may form the operator, R(λ, A) = (λ − A)−1 and since this operator is defined on X and closed (by the closedness of A), the closed graph theorem implies its boundedness. Lastly, R(λ, A) is called the resolvent operator of A at the point λ. 4 We will return later to the spectral theory of semigroups and their generators, but in fact, this definition is the last necessary prerequisite for giving a complete characterization of C0-semigroup generators. Theorem. (Hille-Yosida for Contraction Semigroups) Given a linear map, A, we can say that A will generate a contraction semigroup on X iff it is closed, densely-defined, and for every λ 2 (0; 1), we have jjR(λ, A)jj ≤ λ−1.