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Subject Mathematics ignloeaos n eirus ni eda ihteex the with and deal rates, we asymptotic it decomposability. semigroups, In of characterisation Banach semigroups. and of and sums operators, direct Moszy´nski for diagonal and Lachowicz by proach nerl of integrals Abstract. C h olo hsatcei odvlpater o direct for theory a develop to is article this of goal The 0 smgop nHletsae aallt h eetap- recent the to parallel spaces Hilbert on -semigroups PRTRSEMIGROUPS OPERATOR ietintegrals, Direct W 1. BAA ..NG C.S. ABRAHAM ∗ agba n 2]o ekmeasurability. weak on [21] and -algebras Introduction 1 C 43,3B0 70,(47D07). 47D06, 35B40, 34K30, 0 smgop,gnrtr,asymptotics. generators, -semigroups, C 0 smgop,wsmo- was -semigroups, usin of questions a n operator and nal ,2]adfunc- and 27] 0, one oper- bounded r ain (PDEs) uations es. riua,Nuss- articular, fdrc inte- direct of t rt sfurther is arity gasde not does egrals yNesnin Nielsen by d rwr within work er n expansion and ia systems. sical sregarding ms spaces, a theorems, ian istence n largely and , mprehensive eue to used be n aigfrom nating reycom- argely c integral ect ost the to ions oe to comes t trs in nterest tof nt f C C 0 0 - - 2 A.C.S. NG treatment on direct sums of C0-semigroups seems to have been produced until 2016 in [26] (even though the use of infinite direct sums was already known as a tool to semigroup theorists). The closest that previous work has come to filling this gap in the literature is the study of semigroups of partial isometries that decompose into direct integrals of truncated shifts (see [17, 18, 30, 34, 42]). This article aims to rectify this situation and properly fill the mostly empty space. Furthermore, we hope to provide an interesting first step towards further developments and applications of direct integral and C0-semigroup theory in the direction of both Cauchy problems and operator algebras. The central results of this article are the following two theorems in Section 4. Theorem 4.1 states that the direct integral of a family of C0-semigroups is itself a C0-semigroup on the direct integral space if and only if the family is uniformly exponentially bounded. Theorem 4.2 states that if a decom- posable operator generates a C0-semigroup, then the individual operators comprising this original decomposable operator also individuallly generate C0-semigroups, the direct integral of which coincides with the original C0- semigroup. In particular, any C0-semigroup on the direct integral space that is generated by a decomposable operator is itself decomposable. This is particularly interesting, given that there is a distinction between so-called maximally defined operators and those defined via a direct integral (see [21] where this distinction is teased out). Sections 2-3 provide the preliminaries and basic operator-theoretic results needed to tackle the central issue, which, as mentioned, is addressed in Sec- tion 4. Sections 5-7 then deal with special cases, examples, and asymptotic questions that arise out of our central results, including a discussion on a theorem by Maniar and Nafiri [29] and how it follows as a simple corollary of our more general quantified asymptotics result. From a C0-semigroup theoretic perspective, our results are already deeply interesting, answering natural fundamental questions and laying the ground- work for applications to Cauchy problems, quantified asymptotic theory, and rates of decay. However, this article may also lead to two other rich areas of application, yet to be explored. Firstly, since every von Neumann algebra on a separable Hilbert space is a direct integral of factors [40], we hope that further progress can be made in the theory of quantum Markov semigroups (QMSs) on von Neumann alge- bras by building on our results for C0-semigroups on direct integral spaces. QMSs, a natural generalisation of classical Markov semigroups that have interested mathematical physicists for almost fifty years, are a certain type of one-parameter operator semigroup on von Neumann algebras, originally arising out of the study of open quantum systems. Recently, there has been an interest in deriving results concerning QMSs and their generators inspired from classical strongly continuous semigroup theory (see [1, 2] for example). The key difference between QMSs and C0-semigroups is that QMSs act on what are essentially spaces of operators (subspaces of B(H), the space of bounded linear operators on a Hilbert space H) whereas C0-semigroups act directly on a Banach space X, leading to the use of different topologies. DIRECT INTEGRALS OF OPERATOR SEMIGROUPS 3
Nonetheless, ideas from C0-semigroup theory evidently can be useful in de- veloping an understanding of QMSs. On the theory of QMSs and their geneators, see [13, 25, 28, 37] to name but a few milestone papers. Secondly, direct integrals, often called fibre integrals, have recently re- appeared independently from the theory of von Neumann algebras, in the area of homogenisation theory. For examples of this, see [7, 9, 12, 14]. This provides a potential avenue for more applications of our results. See also [31, Section 5] for an area of potential overlap between homogenisation theory (and hence direct integral theory) and quantified asymptotics for C0-semigroups. Acknowledgements. The author thanks Charles Batty, David Seifert, and Stuart White for helpful discussions on the topic and production of this article. He is also grateful to the University of Sydney for funding this work through the Barker Graduate Scholarship. A further especially warm thank you goes to the reviewer for their careful reading and constructive feedback because of which this article is much improved.
2. Preliminaries In this article, we use standard notation, denoting the domain, spectrum, and resolvent set of a closed linear operator B acting on a Hilbert space X (assumed always to be complex) by D(B), σ(B), and ρ(B) respectively. The resolvent operator (λ − B)−1, for λ ∈ ρ(B), will usually be denoted by R(λ, B). We say that B is densely defined if D(B) is dense in X. We also write B(X) for the space of bounded linear operators on X. The following proposition is a simple fact about resolvents that will be used in the article, stated without proof and an immediate corollary of the Neumann series expansion. Proposition 2.1. Let D be a dense subset of E ⊂ C. Suppose that there exists M > 0 such that for all λ ∈ D, λ ∈ ρ(B) and kR(λ, B)k≤ M. Then E ⊂ ρ(B) and kR(λ, B)k≤ M for all λ ∈ E.
We will also use 1G to denote the characteristic function for a set G and the abbreviation ‘a.e.’ to mean either ‘almost every’ or ‘almost everywhere’, depending on the context. Turning now to direct integrals of Hilbert spaces, let Ω be a locally com- pact Hausdorff topological space and µ a σ-finite positive Borel measure on Ω. An important example of this is the counting measure on N with the discrete topology. A Hilbert (space) bundle over base space Ω is a pair (H,π) where π : H→ Ω is a surjection such that for all s ∈ Ω, the fiber of H at s, defined by −1 Hs = π (s), is a Hilbert space with an inner product h·, ·is and induced norm k · ks. A section x of H is a function x :Ω →H such that π ◦ x(s)= s for all s ∈ Ω. Henceforth, π will be notationally supressed and the Hilbert space bundle will be referred to by H alone. A measurable field of Hilbert spaces is a pair (H, F) where H is a Hilbert bundle and F is a collection of sections satisfying the following conditions:
(1) For all x,y ∈ F, the C-valued function s → hx(s),y(s)is is µ- measurable; 4 A.C.S. NG
(2) If z is a section such that for all x ∈ F, the function s → hx(s), z(s)is is µ-measurable, then z ∈ F; and ∞ (3) There exists a sequence {ξi}i=1 ⊂ F called a fundamental sequence ∞ for H such that for all s ∈ Ω, the elements {ξi(s)}i=1 are dense in Hs. F is called a choice of measurable sections of H. Given a measurable field of Hilbert spaces (H, F) with associated measure µ, the direct integral ⊕ Hs dµ(s) ZΩ is the Hilbert space of all measurable sections x ∈ F such that
2 2 kxk = kx(s)ks dµ(s) < ∞ ZΩ modulo measurable sections that are 0 µ-a.e. and equipped with the inner product
hx,yi = hx(s),y(s)is dµ(s). ZΩ We state some simple properties without proof (see [20, Section 3] and [43, Lemma 35]). Proposition 2.2. Let µ and Ω be as above. The following statements are true. ∞ (i) If M = {Mk}k=1 is a countable collection of sets with finite measure ∞ such that Ω= k=1 Mk, the triple sequence given by
ξi,j,kS (·)= ξi(·)1{s∈Ω:kξi(s)ks≤j}(·)1Mk (·) is also a fundamental sequence after rearrangement into a sequence of one parameter. Furthermore, this fundamental sequence has elements ⊕ that are pointwise bounded and have finite Ω Hs dµ(s) norm. Hence, from now on we can assume that the fundamental sequence is pointwise ⊕ R bounded and contained in Ω Hs dµ(s). (ii) There exists a sequence {e }∞ of measurable sections such that for Ri i=1 every s ∈ Ω, the vectors ei(s) form an orthonormal basis for the fiber Hs.
(iii) {1Mk ei : i, k ∈ N} is a countable subset such that its span is dense in ⊕ ⊕ Ω Hs dµ(s). Hence, Ω Hs dµ(s) is separable. R ⊕ R ′ ′ Given H = Ω Hs dµ(s), if Ω ⊂ Ω is measurable and µ(Ω ) > 0, then we ⊕ can form the direct integral ′ H dµ(s) using the same definition as before, R Ω s ′ ⊕ with Ω replacing Ω. ′ H dµ(s) can be identified with the subspace of H Ω sR given by R⊕ ′ x ∈ Hs dµ(s) : x(s) = 0 for a.e. s∈ / Ω . ZΩ ⊕ ∞ ⊕ Ω′ Hs dµ(s) is closed, since if {xn}n=1 ⊂ Ω′ Hs dµ(s) and xn → x in H, then there exists a subsequence such that xnk (s) → x(s) in Hs for a.e. s ∈ Ω. R ′ R ⊕ In particular, x(s) = 0 for a.e. s∈ / Ω . Hence x ∈ Ω′ Hs dµ(s). R DIRECT INTEGRALS OF OPERATOR SEMIGROUPS 5
3. Direct Integrals of Operators In most discussions on direct integrals, for a given measurable field (H, F), a family of bounded operators {T (s) ∈ B(Hs) : s ∈ Ω} forms a measurable field of operators if the function
s 7→ hT (s)x(s),y(s)is is measurable for all x,y ∈ F. Some treatments also require that
s 7→ kT (s)kB(Hs) be essentially bounded. When this extra condition is satisfied, we shall call {T (s) ∈ B(Hs) : s ∈ Ω} a bounded measurable field of operators. We will not go into a discussion of measurable fields and direct integrals of bounded operators, these can be found in [15, 32, 39]. The standard generalisation to closed operators, first given by Nussbaum [33], is through the characteristic matrices introduced by Stone [38]. The characteristic matrix (Pi,j) of a closed operator A on a Hilbert space H is the 2 × 2 matrix of bounded operators representing the projection P of H × H onto the closed subspace Γ(A), the graph of A. Nussbaum defined a measurable field of closed operators to be a family of closed operators {A(s) : D(A(s)) ⊂ Hs → Hs : s ∈ Ω} such that for each i, j, the family {Pi,j(s) : s ∈ Ω} forms a measurable field of bounded operators. This is shown to be consistent with the bounded case in [33, Proposition 6] We provide an alternative definition which turns out to be equivalent modulo a resolvent condition. We define a measurable field of closed opera- tors to be a family of closed operators {A(s) : D(A(s)) ⊂ Hs → Hs : s ∈ Ω} satisfying the requirement that there exists a fixed ν ∈ C such that for a.e. s ∈ Ω, ν ∈ ρ(A(s)) and the family of resolvents {R(ν, A(s)) : s ∈ Ω} adjusted on a set of measure zero is a bounded measurable field of operators. When there exists such a ν ∈ C, the equivalence of definitions follows from the simple fact that adding scalar multiples of the identity does not change measurability in either definition and that inverses of Nussbaum measurable fields are measurable when they exist because the characteristic matrix of the inverse is merely a rearrangement of the components (see [33, Proposition 5]). Note that since R(λ, B) = (λ − ν + R(ν,B)−1)−1, (λ, ν ∈ ρ(B)), for any operator B, {R(ν, A(s)) : s ∈ Ω} is Nussbaum measurable if and only if {R(λ, A(s)) : s ∈ Ω} is Nussbaum measurable for all λ, ν ∈ ρ(A(s)) for a.e. s ∈ Ω. We can now define the direct integral of a measurable field of closed op- erators {A(s) : D(A(s)) ⊂ Hs → Hs : s ∈ Ω} on the maximal reasonable ⊕ domain in Ω Hs dµ(s) as the operator A given by R ⊕ D(A)= x ∈ Hs dµ(s) : x(s) ∈ D(A(s)) for a.e. s ∈ Ω, ZΩ ⊕ A(·)x(·) ∈ Hs dµ(s) , ZΩ A = A(·)x(·), x ∈ D(A), 6 A.C.S. NG
⊕ and denoted by Ω A(s) dµ(s). In accordance with this definition, if we write A = ⊕ A(s) dµ(s), there is R Ω an implicit assumption that there exists at least one λ ∈ C such that for a.e. R s ∈ Ω λ ∈ ρ(A(s)) and that s 7→ kR(λ, A(s))kB(Hs ) is essentially bounded. ⊕ ′ ′ Given A = Ω A(s) dµ(s), if Ω ⊂ Ω is measurable and µ(Ω ) > 0, then ⊕ we can form the direct integral of closed operators A ′ = ′ A(s) dµ(s) on R Ω Ω ⊕ ′ Ω′ Hs dµ(s) using the same definitions as before, with Ω replacing Ω. When ⊕ ⊕ R ′ considering Ω′ Hs dµ(s) as a closed subspace of Ω Hs dµ(s), AΩ coincides R ⊕ with the restriction of A to ′ H dµ(s), that is, R Ω s R ⊕ R ′ ′ D(AΩ )= Hs dµ(s) ∩ D(A) and AΩ = A|D(A ′ ). ′ Ω ZΩ Note that there may exist operators A on a direct integral space H = ⊕ Ω Hs dµ(s) that are not themselves, the direct integral of a measurable field of closed operators. To see this, take Ω = {1, 2} with the discrete R 2 measure and (H,π) to be the bundle given by H1 = H2 = C. Then C can be considered as the bundle H. The linear operators on H are the 2 × 2 matrices and only the diagonal matrices are direct integrals of operators on ⊕ H1 and H2. In the case, however, where A = Ω A(s) dµ(s) for a measurable field of closed operators {A(s) : D(A(s)) ⊂ Hs → Hs : s ∈ Ω}, A is said to be decomposable. Decomposable operators areR characterised in [33, Corollary 4] as those closed operators that commute with every so-called bounded diagonalisable operator. We now provide some basic properties of direct integrals of operators. First, we deal with boundedness.
⊕ Proposition 3.1. Let A = Ω A(s) dµ(s) be densely defined and M ≥ 0. The following are equivalent: R (i) A(s) ∈B(Hs) for a.e. s ∈ Ω and ess-supkA(s)kB(Hs) ≤ M. s∈Ω ⊕ ⊕ ⊕ (ii) Ω A(s) dµ(s) ∈B Ω Hs dµ(s) and k Ω A(s) dµ(s)k≤ M. R R R ⊕ Note that s 7→ kA(s)kB(Hs) is measurable since Ω Hs dµ(s) is separable so that kA(s)kB(Hs) is a supremum of countably many measurable functions of 1/2 R the form hA(s)x(s), A(s)x(s)is . Proof. Assume (i). Then for any x ∈ D(A),
2 2 2 2 2 kAxk = kA(s)x(s)ks dµ(s) ≤ M kx(s)ks dµ(s) = (Mkxk) , ZΩ ZΩ and we are done by density of D(A). For the converse, we must take care that any offending function we con- ∞ struct is measurable. Let Q = {a + bi : a, b ∈ Q}. Taking the set {ei}i=1 from Proposition 2.2,
n aiei : ai ∈ Q,n ∈ N ( ) Xi=1 DIRECT INTEGRALS OF OPERATOR SEMIGROUPS 7
∞ ∞ is a countable set which we relabel as {fi}i=1 such that {fi(s)}i=1 is dense in Hs for all s ∈ Ω. Let
Ki := {s ∈ Ω : kA(s)fi(s)k > Mkfi(s)ks} , (i ∈ N).
These sets are clearly measurable. If for a.e. s ∈ Ω, kA(s)fi(s)k≤ Mkfi(s)ks for all i ∈ N, then kA(s)k≤ M for a.e. s. Thus, if (i) fails, there exists j ∈ N such that µ(Kj) > 0. ′ ⊕ Let x (·) := fj(·)1Kj (·) and note that it is measurable and in Ω Hs dµ(s). Since R ′ 2 2 2 2 ′ 2 kAx k = kA(s)fj(s)ks dµ(s) > M kfj(s)ks dµ(s) = (Mkx k) , ZKj ZKj (ii) also fails.
Second, we state a theorem on adjoints and inverses.
⊕ Theorem 3.2. Suppose that A = Ω A(s) dµ(s). Then the following are true. R (i) A∗ exists if and only if A(s)∗ exists a.e., in which case
⊕ A∗ = A(s)∗ dµ(s). ZΩ (ii) A−1 exists (as a bounded operator) if and only if A(s)−1 exists and is essentially bounded, in which case ⊕ A−1 = A(s)−1 dµ(s). ZΩ Proof. [10, Proposition 3.5] or [33, Theorem 3] proves this theorem for the Nussbaum case. Since we have the assumption of boundedness in (ii), all that remains to show is that the resolvent conditions also hold for (i). This follows easily from the fact that ρ(B∗) = {λ : λ ∈ ρ(B)} and R(λ, B∗) = R(λ, B)∗ for any unbounded densely defined operator B.
⊕ Note that this also implies that if A = Ω A(s) dµ(s), A is densely defined if and only if D(A(s)) is densely defined for a.e. s ∈ Ω since an operator B is densely defined if and only if B∗ exists.R Some basic spectral properties of direct integrals of operators follow im- mediately from this.
⊕ Corollary 3.3. Suppose that A = Ω A(s) dµ(s). The following statements are true. R (i) ρ(A)= λ ∈ C : λ ∈ ρ(A(s)) for a.e. s ∈ Ω n and ess-supkR(λ, A(s))kB(Hs ) < ∞ s∈Ω and o ⊕ R(λ, A)= R(λ, A(s)) dµ(s), (λ ∈ ρ(A)). ZΩ 8 A.C.S. NG
(ii) σ(A)= λ ∈ C : there exists G ⊂ Ω n such that µ(G) > 0 and λ ∈ σ(A(s)) for all s ∈ G o ∪ λ ∈ C : λ ∈ ρ(A(s)) for a.e. s ∈ Ω n and ess-supkR(λ, A(s))kB(Hs ) = ∞ . s∈Ω o This corollary leads to another corollary, covering and generalising the counting measure case for Hilbert spaces in [26, Corollary 3.7], following its proof. ⊕ Corollary 3.4. Suppose that A = Ω A(s) dµ(s) and that there exists δ > 0 satisfying the following condition: If G ⊂ Ω with µ(G) > 0 and λ ∈ ρ(A(s)) R for a.e. s ∈ G, then there exist Gλ ⊂ G with µ(Gλ) > 0 and η ∈ C such that η ∈ σ(R(λ, A(s))) and
|η|≥ δkR(λ, A(s))kB(Hs ), (s ∈ Gλ). Then σ(A) is equal to the closure of λ ∈ C : there exists K ⊂ Ω such that µ(K) > 0 n and λ ∈ σ(A(s)) for all s ∈ K .
Proof. The inclusion ‘⊇’ follows from Corollary 3.3(ii). For theo opposite inclusion let λ ∈ ρ(A(s)) for a.e. s and suppose that
ess-supkR(λ, A(s))kB(Hs ) = ∞. s∈Ω We are done if we show that λ can be approximated by z such that there exists K ⊂ Ω of positive measure with z ∈ σ(A(s)) for all s ∈ K. Let ε > 0. By assumption, there exists G ⊂ Ω of positive measure such 1 that λ ∈ ρ(A(s)) and kR(λ, A(s))ks > δε for all s ∈ G. Hence, there exists Gλ ⊂ G of positive measure and η ∈ C such that η ∈ σ(R(λ, A(s)) for all 1 s ∈ Gλ and |η| > ε . By [3, Proposition B.2], the spectral mapping theorem for the resolvent, there exists z ∈ C for which z ∈ σ(A(s)) for all s ∈ Gλ −1 such that (z − λ) = η. In particular, |z − λ| < ε and µ(Gλ) > 0. It is worth mentioning that Azoff [4] and Chow [10] delve much deeper into the spectral theory of direct integrals. In particular, [4, Examples 4.2, 4.4] show the importance of essential boundedness in Corollary 3.3 by demonstrating that not much can be said about the spectrum of A given only the spectra of a.e. A(s). Finally, we turn to compactness. Before we prove the following lemma, recall that a measurable subset A ⊂ Ω is called an atom (with respect to µ) if µ(A) > 0 and for all measurable B ⊂ A, either µ(B)=0or µ(B)= µ(A). ⊕ Lemma 3.5. If Ω A(s) dµ(s) is compact and kA(s)kB(Hs) ≥ ε for some ε> 0 and a.e. s ∈ Ω, then Ω contains an atom. R Proof. Assume otherwise. Then there exist subsets Ω = G1 ) G2 ) . . . such that µ(Gi) > µ(Gi+1) > 0 for all i ∈ N. Writing Ki = Gi \ Gi+1, we ∞ get that G = i=1 Ki is a disjoint union and µ(Ki)= µ(Gi) − µ(Gi+1) > 0 S DIRECT INTEGRALS OF OPERATOR SEMIGROUPS 9 for all i ∈ N. Using the same construction as in the proof of Proposition ′ ′ 3.1, for all i ∈ N, there exists a measurable subset Ki ⊂ Ki with µ(Ki) > 0 ⊕ ε and a (normalised) function yi ∈ Ω Hs dµ(s) such that kA(s)yi(s)ks > 2 ′ ′ ′ for all s ∈ Ki, kyi(s)ks = 1 for all s ∈ Ki, and yi(s) = 0 for all s∈ / Ki. Let ′ −1/2 R⊕ xi := µ(Ki) yi(s). Then xi ∈ Ω Hs dµ(s) and kxik = 1 for all i ∈ N. But R 2 kAxi−Axjk
′ −1 2 ′ −1 2 = µ(Ki) kA(s)yi(s)ks dµ(s)+ µ(Kj) kA(s)yj(s)ks dµ(s) ZΩ ZΩ ε2 > 2 ′ ′ for all i 6= j as Ki ⊂ Ki and Kj ⊂ Kj are disjoint. Hence, A cannot be compact. ∞ Recall that for a direct sum of operators T = i=1 Ti on a direct sum ∞ of Hilbert spaces Hi, T is compact if and only if Ti is compact for all i=1 L i ∈ N and kTikB(Hi) → 0 (proved via finite-rank approximations). Note also that for Radon measures,L if A ⊂ Ω is an atom, then A is of the form {x}∪N where x ∈ A and µ(N) = 0. ⊕ Theorem 3.6. Let µ be a Radon measure. Suppose that A = Ω A(s) dµ(s). Then A is compact if and only if Ω=Ω ∪ Ω where Ω and Ω are disjoint, 1 2 1 R2 Ω1 = {s1,s2,... } is a countable set of points such that µ(si) > 0 and A(si) is compact and non-zero for all i ∈ N, kA(si)kB(Hs) → 0 as i → ∞, and A(s) = 0 for a.e. s ∈ Ω2. Proof. The ‘if’ direction is simply the case of a direct sum of compact op- ∞ ∞ erators tailing to zero, rewriting A = i=1 A(si) on i=1 Hsi . There the direct sum can clearly be approximated by operators of finite rank. L L For the ‘only if’ direction, let Ω1 be the set of points of positive measure ∞ and let {Mk}k=1 be the collection of sets in Proposition 2.2 of finite measure exhausting Ω. Then ∞ ∞ µ(M ) S = s ∈ S ∩ M : µ ({s}) ≥ k . k n k[=1 n[=1 Now µ(M ) s ∈ S ∩ M : µ ({s}) ≥ k ≤ n, (k,n ∈ N), k n so that S is the countable union of finite sets. Writing S = {s1,s2,... }, it ⊕ ⊕ is clear that the restriction S A(s) dµ(s) of A to the subspace S Hs dµ(s) is also compact and that we can write R R ⊕ ∞ ⊕ ∞
Hs dµ(s)= Hsi and A(s) dµ(s)= A(si). S S Z Mi=1 Z Mi=1 It follows from the direct sum case that A(si) is compact for all i ∈ N and kA(s )k → 0 as i →∞. i B(Hsi ) If A(s) = 0 for a.e. s ∈ Ω \ S, we are done. If not, there exists G ⊂ Ω \ S such that µ(G) > 0 and kA(s)kB(Hs) > 0 for all s ∈ G. Without loss of 10 A.C.S. NG
generality, we can assume by taking a smaller subset, that kA(s)kB(Hs) ≥ ε for some ε > 0 and a.e. s ∈ G. Applying Lemma 3.5 to the subspace ⊕ G Hs dµ(s), we can add another point of positive measure s0 ∈ Ω \ S to S. This contradicts that S is the set of all points of positive measure and we areR done.
Thus, Theorem 3.6 tells us that the only compact direct integrals of op- erators are, in fact, direct sums.
4. Direct Integrals of C0-Semigroups In this section, we begin by proving that we can take direct integrals of C0-semigroups that have a uniform exponential growth bound to get a C0- semigroup on the direct integral space. Furthermore, the generator of the direct integral C0-semigroup is the direct integral of the generators. (·) Theorem 4.1. Let T (s) : R+ → B(Hs) : s ∈ Ω be a collection of C0- semigroups. If for each t ≥ 0, the direct integral of {T t(s) : s ∈ Ω} exists as ⊕ a bounded operator, then T : R+ →B Ω Hs dµ(s) defined and denoted by
⊕R T (t) := T t(s) dµ(s) ZΩ is a C0-semigroup if and only if there exist M ≥ 1,ω ∈ R such that t ωt kT (s)kB(Hs) ≤ Me for a.e. s ∈ Ω and all t> 0. In this case, kT (t)k ≤ Meωt for all t ≥ 0 and its generator A is given by ⊕ A = A(s) dµ(s) ZΩ where A(s) is the generator of T t(s) for a.e. s ∈ Ω.
⊕ Proof. First, we prove the ‘if’ direction. T : R+ → B( Ω Hs dµ(s)) clearly satisfies the semigroup property. We now check the strong continuity at 0. (·) R ⊕ Since T (s) is a C0-semigroup for each s ∈ Ω, for any x ∈ Ω Hs dµ(s), t T (s)x(s) → x(s) in Hs as t → 0. Hence we have pointwise convergence of t R kT (s)x(s) − x(s)ks → 0. However,
t 2 t 2 ωt 2 2 kT (s)x(s) − x(s)ks ≤ kT (s)x(s)ks + kx(s)ks ≤ (Me + 1) kx(s)ks ⊕ for all t ≥ 0. Since x ∈ Ω Hs dµ(s), by Lebesgue’s dominated convergence theorem, R t 2 kT (s)x(s) − x(s)ks dµ(s) → 0. ZΩ ωt By Proposition 3.1, kT (t)k≤ Me for all t ≥ 0. Since any C0-semigroup has an exponential bound, Proposition 3.1 also proves the ‘only if’ direction. Finally, let B be the generator of the direct integral C0-semigroup T . Our goal is to prove that B = A, but we must first check that A exists as a direct DIRECT INTEGRALS OF OPERATOR SEMIGROUPS 11 integral. Let λ = ω + 1. Then by the Laplace transform representation for resolvents of generators, λ ∈ ρ(A(s)) for a.e. s and ∞ −λt t hR(λ, A(s))x(s),y(s)is = e T (s)x(s) dt,y(s) 0 s Z∞ −λt t = e hT (s)x(s),y(s)is dt Z0 for all x,y ∈ F. Since the final integrand is measurable by assumption, Fubini’s theorem tell us that {R(λ, A(s)) : s ∈ Ω} forms a measurable field of bounded operators. Hence {A(s) : s ∈ Ω} is a measurable field of closed operators. We now show that B = A. Again, let λ = ω +1. Then λ ∈ ρ(B), λ ∈ ρ(A(s)) and for some fixed C > 0, kR(λ, B)k, kR(λ, A(s))k ≤ C for a.e. s ∈ Ω. By Proposition 3.3, λ ∈ ρ(A). Once again using the Laplace transform representation for resolvents of generators and Corollary 3.3(i), we have
∞ ∞ ⊕ (4.1) R(λ, B)x = e−λtT (t)x dt = e−λtT t(s) dµ(s) x dt, 0 0 Ω Z ⊕ Z Z R(λ, A)x = R(λ, A(s)) dµ(s) x, ZΩ and for a.e. s ∈ Ω, ∞ (4.2) R(λ, A(s))x(s)= e−λtT t(s)x(s) dt. Z0 ⊕ The outer integral in Equation (4.1) is a Bochner integral in Ω Hs dµ(s) and the integral in Equation (4.2) is a Bochner integral in Hs. Let x,y ∈ ⊕ R Ω Hs dµ(s) be arbitrary. Then R ∞ ⊕ e−λtT t(s) dµ(s) xdt,y 0 Ω Z Z ∞ = e−λtT t(s)x(s),y(s) dµ(s)dt 0 Ω s Z Z∞ D E (4.3) = e−λtT t(s)x(s),y(s) dtdµ(s) Ω 0 s Z Z ∞D E = e−λtT t(s)x(s) dt,y(s) dµ(s) ZΩ Z0 s
= R(λ, A(s))x(s),y(s) s dµ(s) ZΩ ⊕ = R(λ, A(s)) dµ(s) x,y , ZΩ where Equation (4.3) holds due to Fubini’s theorem. Hence R(λ, B)x,y = R(λ, A)x,y for arbitrary x and y, so that R(λ, B) = R(λ, A) and in par- ticular, B = A.
12 A.C.S. NG
A natural question to ask is whether the sufficient conditions for a direct integral of operators to generate a C0-semigroup are also necessary condi- tions (see [26, Theorem 4.4]). A positive answer for this question is given in the following theorem. ⊕ Theorem 4.2. If A = Ω A(s) dµ(s) generates a C0-semigroup T (·) on ⊕ H dµ(s), then A(s) generates a C -semigroup denoted T (·)(s) on H for Ω s R 0 s a.e. s ∈ Ω which has the same exponential bounds as the one generated by AR . Furthermore, ⊕ T (t)= T t(s) dµ(s). ZΩ ⊕ Proof. If A generates a C0-semigroup on Ω Hs dµ(s), then there exist M ≥ 1,ω ∈ R such that kT (t)k≤ Meωt for all t ≥ 0 and for every λ>ω, one has R λ ∈ ρ(A) and k[(λ − ω)R(λ, A)]nk≤ M for all n ∈ N. By Propositions 3.3(2) and 3.1, for each λ>ω and n ∈ N, there exists a measurable Uλ,n ⊂ Ω with µ(Ω \ Uλ,n) = 0 such that λ ∈ ρ(A(s)) and n k[(λ − ω)R(λ, A(s))] ks ≤ M for all s ∈ Ω\Uλ,n. Let G = λ∈Qω+,n∈N Uλ,n where Qω+ = {µ ∈ Q : µ>ω}. Then µ(Ω \ G) = 0 and for all s ∈ G, we have Q ⊂ ρ(A(s)) and S ω+ n (4.4) k[(λ − ω)R(λ, A(s))] ks ≤ M, (λ ∈ Qω+,n ∈ N).
Clearly Qω+ is dense in {µ ∈ R : µ>ω} so by considering bounded subsets of {µ ∈ R : µ>ω} and applying Proposition 2.1 to the relation (4.4), we get that for all λ ∈ R with λ>ω, we have λ ∈ ρ(A(s)) and by continuity, (4.4) holds for all s ∈ G. It follows that for all s ∈ G and hence t for a.e. s ∈ Ω, A(s) generates a C0-semigroup T (s) such that t ωt kT (s)kB(Hs) ≤ Me for all t ≥ 0. Let Re λ>ω so that λ ∈ ρ(A(s)) for a.e. s ∈ Ω. Since {A(s) : s ∈ Ω} is a measurable field of closed operators, {R(λ, A(s)) : s ∈ Ω} forms a measurable field of bounded operators. Hence for any x,y ∈ F, the Laplace transform representation for resolvents of generators gives us that ∞ −λt t s 7→ R(λ, A(s))x(s),y(s) s = e T (s)x(s) dt,y(s) Z0 s ∞ −λt t = e T (s)x(s),y(s) s dt Z0 is measurable. Let ∞ −λt t F (λ, s) := e T (s)x(s) − x(s),y(s) s dt, Z0 which is measurable by Fubini’s theorem. Notice that for fixed s, F (λ, s) is t the Laplace transform with respect to t of g(t,s)= T (s)x(s) − x(s),y(s) s
DIRECT INTEGRALS OF OPERATOR SEMIGROUPS 13 which satisfies g(0,s) = 0, is continuous in t by strong continuity of T t(s), and t |g(t,s)| = T (s)x(s) − x(s),y(s) s ωt ω′t ≤ ( ce + 1)kx(s)ksky(s)k s ≤ (c + 1)kx(s)ksky(s)kse where ω′ = max {ω, 0} so that g is exponentially bounded with respect to t. Applying Fubini’s theorem two more times, we get that ′ 1 u 1 ω +1+ir s 7→ eλtF (λ, s) dλdr u 2π ′ Z0 Zω +1−ir is also measurable for each t ≥ 0. By [3, Theorem 4.2.21(b)], a Tauberian theorem, ′ 1 u 1 ω +1+ir g(t,s) = lim eλtF (λ, s) dλdr u→∞ u 2π ′ Z0 Zω +1−ir t for a.e. s ∈ Ω. Hence s 7→ T (s)x(s) − x(s),y(s) s is the pointwise a.e. limit of measurable functions for each t ≥ 0. This together with the uniform t ωt t exponential bounds kT (s)kB(Hs) ≤ Me implies that {T (s) : s ∈ Ω} form a measurable field of bounded operators for each t ≥ 0. ⊕ t Theorem 4.1 then implies that A generates Ω T (s) dµ(s), hence ⊕ R T (t)= T t(s) dµ(s). ZΩ
As mentioned in the introduction, Theorem 4.2 says that decomposable generators generate C0-semigroups that are decomposable into individual C0-semigroups.
Remark 4.3. If A is a decomposable operator that generates a C0-semigroup T (·), the following argument is an alternative way via the characterisation by Nussbaum to see that for each t ≥ 0, T (t) is a decomposable operator. Since A is decomposable, Corollary 3.3(i) implies that R(λ, A) is decom- posable for all λ ∈ ρ(A). In particular, R(λ, A) commutes with all bounded diagonalisable operators [33, Corollary 4] and hence by [3, Proposition 3.1.5], T (t) commutes with all bounded diagonalisable operators for any t ≥ 0. The characteristaion [33, Corollary 4] again implies that T (t) is decomposable for all t ≥ 0. However, unlike Theorem 4.2 this argument does not guaran- tee that T (·) decomposes into a family of C0-semigroups, since all we would know from this is that T (t) is decomposable for each t ≥ 0. Difficulties arise when trying to show the existence of a semigroup on Hs for any particular s. We also have the following corollary which generalises [26, Theorem 4.4] in the Hilbert space case in a weaker way than Theorem 4.2. ⊕ Corollary 4.4. Suppose that A = Ω A(s) dµ(s) generates a C0-semigroup ⊕ ′ on Ω Hs dµ(s). Then for every measurable subset Ω ⊂ Ω with positive ⊕ R measure, Ω′ A(s) dµ(s) generates a C0-semigroup on the closed subspace ⊕ R ′ H dµ(s). Ω s R R 14 A.C.S. NG
Proof. Theorem 4.2 gives the pointwise and uniform a.e. conditions for {A(s) : s ∈ Ω′} necessary to apply Theorem 4.1 with Ω′ replacing Ω. Remark 4.5. A proof of this corollary can be obtained in an analogous way as ⊕ the proof for [26, Theorem 4.4]. The key step was to show that Ω′ Hs dµ(s) is ⊕ invariant under T , the C -semigroup generated by A. Let x ∈ ′ H dµ(s). 0 R Ω s By the Post-Widder inversion formula [19, Chapter III Corollary 5.5], R n n n ⊕ n n n T (t)x = lim R , A x = lim R , A(s) dµ(s) x n→∞ t t n→∞ Ω t t Z for any t ≥ 0 where the resolvents in the integrand of the direct integral exist for a.e. s ∈ Ω. By definition of the direct integral of operators, ⊕ n n n R , A(s) dµ(s) x (s) = 0 Ω t t Z ′ ⊕ for a.e. s∈ / Ω . Hence T (t)x is the limit in Ω Hs dµ(s) of elements in ⊕ ⊕ ′ H dµ(s) and by closedness, is in ′ H dµ(s). We can apply [19, Chapter Ω s Ω s R II Proposition 2.3] to complete the proof. R However, this method could notR be directly used to prove Theorem 4.2 ⊕ since Hs does not embed into Ω Hs dµ(s) in any meaningful way and hence, like in Remark 4.3, difficulties potentially arise as to the existence of a R semigroup for all t ≥ 0 on Hs for any particular s.
5. Direct Integrals of Special Classes of Semigroups A natural question to ask is whether there are similar results to Theorems 4.1 and 4.2 for special classes of semigroups (see [19, Chapter II Section 4]). Ahead of the following discussion, we note that these classes can be classified through spectral conditions in such a way that their direct integral theory once again, like in the general case, reduces to a.e. uniform conditions on the individual fibres.
5.1. Bounded Analytic Semigroups. First, we provide an easy conse- quence of Proposition 3.3 that deals with sectoriality which we now define. A closed linear operator A with dense domain D(A) is sectorial of angle π δ ∈ [0, 2 ) if π (1) Σπ/2+δ ⊂ ρ(A), where Σπ/2+δ = λ ∈ C : | arg λ| < 2 + δ \{0}; and (2) For each ε ∈ (0, δ), there exists M ≥ 1 such that ε M kR(λ, A)k≤ ε , (0 6= λ ∈ Σ ). |λ| π/2+δ−ε ⊕ Proposition 5.1. Suppose that A = Ω A(s) dµ(s). Then A is sectorial of angle δ ∈ [0, π ) if and only if Σ \ {0} ⊂ ρ(A(s)) and for all ε ∈ (0, δ) 2 π/2+δ R there exists Mε ≥ 1 such that M kR(λ, A(s))k ≤ ε , (λ ∈ Σ \ {0}), s |λ| π/2+δ−ε for a.e. s ∈ Ω. DIRECT INTEGRALS OF OPERATOR SEMIGROUPS 15
Proof. The ‘if’ direction follows immediately from Corollary 3.3(i) followed by Proposition 3.1 as the uniform resolvent bound for a.e. s given λ ∈ Σπ/2+δ guarantees that such λ ∈ ρ(A). Now let Q = {a + ib ∈ C : a, b ∈ Q}. The ‘only if’ direction is then proved by showing, in the same way as the proof for Theorem 4.2, that Σπ/2+δ ∩Q\{0}⊂ ρ(A(s)) and that for each ε ∈ (0, δ), the same constant Mε for bounding kλR(λ, A)k also bounds kλR(λ, A(s))ks for all λ ∈ Σπ/2+δ−ε ∩ Q \{0} for a.e. s ∈ Ω. Again, as in the proof of Theorem 4.2, we are done by Proposition 2.1 and continuity of the resolvent applied to compact subsets of the sectors. π In other words, A is sectorial of angle δ ∈ [0, 2 ) if and only if A(s) is uniformly a.e. sectorial of angle δ for a.e. s. We restate this in terms of bounded analytic semigroups without proof (see [19, Chapter II Section 4a.] and [24, Chapter 3]). ⊕ Theorem 5.2. Suppose that A = Ω A(s) dµ(s). Then A generates a bounded analytic semigroup T of angle δ ∈ [0, π ) if and only if A(s) gener- R 2 ates an analytic semigroup T (·)(s) of angle δ such that for every 0 < δ′ < δ, there exists Mδ′ ≥ 1 with z kT (s)ks ≤ Mδ′ , (z ∈ Σδ′ ), uniformly for a.e. s ∈ Ω. These results can also be extended for general unbounded semigroups (see [24, Propositions 3.16, 3.18, Theorem 3.19]). An analogous version of Proposition 5.1 is also true for operators A whose resolvent set contain sectors minus {0} with angle smaller than π/2 and whose resolvent operators satisfy analogous bounds on these sectors. Our more restricted definition of sectoriality is due to the C0-semigroup setting we are working in (see [19, Chapter II, Definition 4.1]). 5.2. Eventually Differentiable Semigroups. In light of the characteri- sation of eventually differentiable semigroups found in [19, Chapter II The- orem 4.14], a similar result to Theorem 5.2 can be given for eventually differentiable semigroups. ⊕ Theorem 5.3. Suppose that A = Ω A(s) dµ(s). Then A generates an eventually differentiable semigroup exponentially bounded by Meωt for ω ∈ R R if and only if A(s) generates a C0-semigroup also exponentially bounded by Meωt and there exist constants a, b, C > 0 with Θ := λ ∈ C : ae−b Re λ ≤ | Im λ| ⊂ ρ(A(s)) and kR(λ, A(s))ks ≤ C| Im λ| for all λ ∈ Θ with Re λ ≤ ω uniformly for a.e. s ∈ Ω. Proof. The proof is almost exactly the same as that of Proposition 5.1. However, for the ‘if’ direction, we must also check that
ess-supkR(λ, A(s))kB(Hs ) < ∞ s∈Ω 16 A.C.S. NG holds for all λ ∈ Θ. For Re λ ≤ ω, the uniform a.e. resolvent bound is given by C| Im λ|. For Re λ>ω, M kR(λ, A(s))k ≤ s Re λ − ω for a.e. s ∈ Ω by virtue of standard semigroup generator properties.
In other words, the direct integral operator generates an eventually dif- ferentiable semigroup if individual A(s) are eventually differentiable in a uniform a.e. way.
5.3. Immediately Norm-Continuous Semigroups. We introduce the natural notion of norm-continuous in t uniformly a.e. on Ω in order to for- mulate the most obvious result concerning immediately norm-continuous t semigroups. We say that {T (s) : s ∈ Ω} is norm-continuous for t>t0 uniformly a.e. on Ω if for every t>t0 and ε > 0, there exists δ > 0 such that t t1 kT (s) − T (s)kB(Hs) < ε, (|t − t1| < δ), for a.e. s ∈ Ω. Note that δ is independent of s. ⊕ Proposition 5.4. Suppose that A = Ω A(s) dµ(s). Assume that A(s) gen- erates a semigroup T (·)(s) for a.e. s ∈ Ω that are uniformly a.e. exponentially t R bounded and that {T (s) : s ∈ Ω} is norm-continuous for t>t0 uniformly a.e. on Ω. Then A generates an eventually norm-continuous semigroup that is norm-continuous for t>t0. Proof. The direct integral semigroup T (·) exists by Theorem 4.1. Let t > ⊕ t0,ε > 0, and x ∈ Ω Hs dµ(s). Then by norm-continuity for t>t0 on Ω a.e., there exists δ > 0 such that R t t1 2 t t1 2 k(T − T )xk = k(T (s) − T (s))x(s)ks dµ(s) ZΩ t t1 2 2 ≤ kT (s) − T (s)kskx(s)ks dµ(s) ZΩ < ε2kxk2
t t1 t for all |t − t1| < δ. Hence kT − T k < ε for all |t − t1| < δ, proving that T is norm-continuous for t>t0. [19, Chapter II Theorem 4.20] gives us a useful characterisation of imme- diately norm-continuous semigroups on Hilbert spaces, which we can use to prove the following result. ⊕ Theorem 5.5. Suppose that A = Ω A(s) dµ(s). Then A generates an immediately norm-continuous semigroup exponentially bounded by Me−εt R (·) for some ε > 0 if and only if A(s) generates a C0-semigroup T (s) also exponentially bounded by Me−εt and
lim kR(ir, A(s))kB(H ) = 0 r→±∞ s uniformly for a.e. s ∈ Ω. DIRECT INTEGRALS OF OPERATOR SEMIGROUPS 17
Proof. For the ‘if’ direction, Theorem 4.1 ensures that A generates a C0- semigroup that is exponentially bounded by Me−εt. By Proposition 3.1, the fact that lim kR(ir, A)k = 0, r→±∞ and [19, Chapter II Theorem 4.20], A generates an immediately norm- continuous semigroup. For the converse, it is enough to see that as in the proof of Theorem 4.2, there is a set G ⊂ Ω with µ(Ω \ G) = 0 such that for all s ∈ G, iQ ⊂ ρ(A(s)) and kR(ir, A(s))ks ≤ kR(ir, A)k for all r ∈ Q. Continuity of the resolvent and density of iQ in iR completes the proof. 5.4. Immediately Compact Semigroups. [19, Chapter II Theorem 4.29] tells us that a C0-semigroup T is immediately compact if and only if T is immediately norm-continuous and its generator A has compact resolvent. For an operator B, the resolvent R(λ, B) for any λ ∈ ρ(B) cannot be zero. Hence, in light of Theorem 3.6, the only possible setting for a direct integral of semigroups to be immediately compact when µ is a Radon measure is the direct sum case. Combining [19, Chapter II Theorem 4.29] with Theorem 3.6 and Theorem 5.5, we get the following result with trivial proof. Theorem 5.6. Let µ be the counting measure on N and suppose that A = N A(n) dµ(n). Then A generates an immediately compact semigroup if and only if lim kR(ir, A(n))kB(H ) = 0 uniformly for all n ∈ N and there exists R r→±∞ n λ ∈ C such that λ ∈ ρ(A(n)) and R(λ, A(n)) is compact for all n ∈ N and lim kR(λ, A(n))kB(H ) → 0. In particular, if A generates an immediately n→∞ n compact semigroup, then so does A(n) for all n ∈ N.
6. Examples and Applications In this section, we discuss two examples where Theorem 4.1 can be ap- plied. Regrettably, these are somewhat special cases, where extra structure allows us to work more easily with the relevant spaces. The point, however, is that examples exist. The first example is as follows. Let Ω = (0, 1) and take µ to be the Lebesgue measure on (0, 1). It is worth informing the reader here that there is some ambiguity in the literature about what measures and measure spaces are used in direct integral theory. Most papers describe µ as a Borel measure on a locally compact topological space, as we have done so in Section 2. This, however, does not mean we are only restricted to working with the Borel σ- algebra, as the known results hold on measure spaces that contain the Borel σ-algebra (see for example [5, 21, 43]). Now define H by fixing Hs to be the constant Hilbert space L2(R), the space of square integrable complex-valued functions on the real line, for all s ∈ (0, 1). For the choice of measurable sections, we take F to be all sections x : (0, 1) →H such that
s 7→ hxs, giL2 2 ⊕ 2 2 is measurable for all g ∈ L (R). In this case, Ω Hs dµ(s)= L ((0, 1); L (R)) = L2((0, 1) × R), the space of square integrable complex-valued functions on R (0, 1) × R. This is also known as the Hilbert space tensor product L2(0, 1) ⊗ L2(R). 18 A.C.S. NG
Define the operator A on L2(R) by D(A)= {f ∈ L2(R) : f absolutely continuous,f ′ ∈ L2(R)} Af = f ′, (f ∈ D(A)). This is widely known to generate the left-shift semigroup T (·) given by (T tf)(r)= f(r+t) for f ∈ L2(R) and r ∈ R (see for example [19, Chapter II Section 2]). Now for each s ∈ (0, 1), define the operator As by D(As)= D(A) and Asf = (A − sI)f for f ∈ D(As). Elementary differential equation the- 2 2 ory shows that {As : D(As) ⊂ L (R) → L (R) : s ∈ (0, 1)} is a measurable field of closed operators (σ(A) = iR so that σ(As) = −s + iR allowing for a common element of the resolvent sets to exist). Now each As gen- (·) −s· (·) 2 erates the re-scaled semigroup Ts := e T on L (R) and the family (·) 2 (·) {Ts : R+ → B(L (R)) : s ∈ (0, 1)} is uniformly bounded by 1 as T is bounded by 1. Futhermore for every t> 0,
−st t −st s 7→ he T xs,ysis = e xs(r + t)ys(r) dr ZR is measurable for all x,y ∈ F. ⊕ Thus by Theorem 4.1, the direct integral B := Ω As dµ(s) generates a C -semigroup S(·) given by St = ⊕ T t dµ(s) for all t ≥ 0. Of course, 0 Ω s R one can also follow the same computation that is done in general terms in the proof of Theorem 4.1 to show thisR directly. In this simple example, B can be thought of as encoding the range of a continuous pertubation of the differentiation operator. Note that it is essential that Ω be bounded below to ensure the re-scaled semigroups are uniformly exponentially bounded. The second example is the case where the family of closed operators {As : D(As) ⊂ Hs → Hs : s ∈ Ω} is a bounded measurable field of operators. In this case, exponentiation yields a family of uniformly bounded norm- continuous C0-semigroups {exp(·As) : R+ → B(Hs) : s ∈ Ω} (in fact, C0- groups). From this point, we can follow two separate routes to obtain a decomposable C0-semigroup on the direct integral space. The first is, of course, Theorem 4.1. The second, more interesting method, is to apply ⊕ functional calculus. We define the direct integral A = Ω As dµ(s) which is also a bounded operator. From here, we can apply the Riesz-Dunford functional calculus to the exponential function exp(t·) forR each t ≥ 0. This is clearly holomorphic on an open neighbourhood containing σ(As) for a.e. s ∈ Ω. Using Gilfeather’s result [22, Theorem 1], we see that ⊕ exp(tA)= exp(tAs) dµ(s). ZΩ We end this section by mentioning the setting where µ is the counting measure. Here, the theory of direct integrals reduces to that of direct sums which is dealt with in [26], where the theory is developed for Banach spaces more generally and not restricted to Hilbert spaces. In particular, an ex- ample of a stochastic particle system is used to show that the direct sum equivalent of Theorem 4.1, [26, Theorem 4.3], automatically upgrades ‘sep- arately weak’ solutions to ‘strong solutions’, resulting in an extra degree of differentiability for certian initial value problems. DIRECT INTEGRALS OF OPERATOR SEMIGROUPS 19
It remains an interesting avenue of exploration to see if there are physical applications of Theorem 4.1 that require more than the scenario where the individual fibres of the Hilbert space bundle are equal to a fixed Hilbert space. It may be that more applications can be found, perhaps in the flavour of the stochastic example in [26] if our theory can be extended to the Banach space setting. In particular, the motivating link between C0-semigroups and Cauchy problems lends itself to the hope that useful applications remain waiting to be found. Note that direct integrals of Banach spaces have been studied under the rubric of randomly normed spaces in [23].
7. Asymptotics of Direct Integral Semigroups We motivate this final section by beginning with and stating a result concerning the stability of a uniformly bounded countable sequence of C0- semigroups on Hilbert spaces from [29] with an added assumption that we believe is necessary.
Theorem 7.1 ([29, Theorem 3.2]). Let {Tn(t) : R+ →B(Hn) : n ∈ N} be a uniformly bounded sequence of C0-semigroups with corresponding generators An such that for all n ∈ N, iR ⊂ ρ(An). Further assume that −1 (7.1) sup kAn kHn < ∞. n∈N Then for a fixed α> 0 the following conditions are equivalent: −α (i) sup |r| kR(ir, An)kHn < ∞. |r|≥1,n∈N 1/α −1 (ii) sup kt Tn(t)An kHn < ∞. t≥0,n∈N The assumption (7.1) is missing from the statement of [29, Theorem 3.2] as found in [29]. It is ambiguous as to whether or not the authors of that pa- per implicitly assume this condition in [29, Theorem 3.2] without explicitly stating it. The introduction, preparatory work in Section 2, and applica- tion in Theorem 4.4 of their paper indicate that they are, in fact, assuming (7.1). However, their statement of [29, Proposition 3.6] seems to indicate otherwise. In any case, we both believe that the authors of [29] merely ac- cidentally left out (7.1) as an assumption in [29, Theorem 3.2] and use the following example to show that the theorem fails if it is not assumed. We drop the subscripts for the norms as the context provides enough clarity.
Example 7.2. Take An = −1/n to be the multiplication operator acting on Hn = C for all n ∈ N. Then An generates the multiplication semigroup −t/n Tn(t)= e on Hn for all n ∈ N which is uniformly bounded by 1. More- −1 over, ir − An = ir + 1/n which has inverse (ir + 1/n) for all r ∈ R,n ∈ N so that iR ⊂ ρ(An) for all n ∈ N. Condition (i) of Theorem 7.1 is satisfied since −α −α −1 −(1+α) |r| kR(ir, An)k = |r| |ir + 1/n| ≤ |r| , (n ∈ N, |r|≥ 1). However condition (ii) of Theorem 7.1 is not satisfied since for t = 1, 1/α −1 −1/n −1 k1 Tn(1)An k = ke nk≥ e n 1/α −1 so that sup kt Tn(t)An k = ∞. t≥0,n∈N 20 A.C.S. NG
Thus, (7.1) cannot be omitted and furthermore, we believe it is just as natural to assume instead the stronger condition
(7.2) sup kR(ir, An)k < ∞, (r ∈ R), n∈N which we will explain after the following theorem which generalises [29, Theorem 3.2] with the correct assumptions. We will prove the theorem using a much simpler argument than the one found in [29], while still using [8, Theorem 2.4] as in [29].
(·) Theorem 7.3. Let {T (s) : R+ → B(Hs) : s ∈ Ω} be a collection of C0- semigroups with generators A(s) such that for each t ≥ 0, the direct integral t ⊕ of {T (s) : s ∈ Ω} exists as a bounded operator on H = Ω Hs dµ(s) and t ess-sup sup kT (s)kHs < ∞. R s∈Ω t≥0 Further assume that iR ⊂ ρ(A(s)) for a.e. s ∈ Ω and
ess-supkR(ir, A(s))kHs < ∞, (r ∈ R). s∈Ω Then the following are equivalent: −α (i) ess-sup sup |r| kR(ir, A(s))kHs < ∞. s∈Ω |r|≥1 1/α t −1 (ii) ess-sup sup kt T (s)A(s) kHs < ∞. s∈Ω t≥0 Proof. Dropping the subscripts on the norms, there exists M > 0 such that kT t(s)k≤ M for all t ≥ 0 and a.e. s. Thus, ⊕ T (t)= T t(s) dµ(s) ZΩ is a C0-semigroup on H bounded by M with generator ⊕ A = A(s) dµ(s) ZΩ by Theorem 4.1. Moreover, iR ⊂ ρ(A) by Corollary 3.3. Assuming (i), we see that (7.3) sup |r|−αkR(ir, A)k < ∞. |r|≥1 Thus by [8, Theorem 2.4], (7.4) sup kt1/αT (t)A−1k < ∞. t≥0 Since ⊕ T (t)A−1 = T t(s)A(s)−1 dµ(s), ZΩ (ii) follows. Now assuming (ii), we similarly have (7.4), which implies (7.3) by [8, Theorem 2.4] once again. Since ⊕ R(ir, A)= R(ir, A(s)) dµ(s), (r ∈ R), ZΩ (i) follows. DIRECT INTEGRALS OF OPERATOR SEMIGROUPS 21
Thus, [29, Theorem 3.2] with the additional assumption (7.2) follows by taking the counting measure. In fact, the only theory necessary in this specific discrete case is that of direct sums, covered in [26]. We also see from the above proof that if we only assume −1 ess-supkA(s) kHs < ∞, (s ∈ Ω), s∈Ω rather than the a.e. uniform resolvent bounds along the whole imaginary axis, we would still have 0 ∈ ρ(A) and hence, in this case, if Theorem 7.3(ii) holds, then (7.4) holds as well. Hence [6, Proposition 1.3] would imply that iR ⊂ ρ(A) and further that
ess-supkR(ir, A(s))kHs < ∞, (r ∈ R), s∈Ω demonstrating that the stronger condition (7.2) is indeed just as natural to assume as (7.1). The method of passing to a direct integral (or sum) in order to obtain the relation between uniform resolvent bounds and ‘uniform strong’ stability can be applied to other quantified Tauberian theorems such as those found in [6, 8, 35] (strongly continuous) and [36] (discrete).
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(A.C.S. Ng) St Edmund Hall, Queen’s Lane, Oxford OX1 4AR, UK E-mail address: [email protected]