Adjoints of Sums of M-Accretive Operators and Applications to Non-Autonomous Evolutionary Equations

: We provide certain compatibility conditions for m-accretive operators such that the adjoint of the sum is given by the closure of the sum of the respective adjoint. We revisit the proof of well-posedness of the abstract class of partial differential-algebraic equations known as evolutionary equations. We show that the general mechanism provided here can be applied to establish well-posedness for non-autonomous evolutionary equations with L ∞ -coefficients thus not only generalising known results but opening up new directions other methods such as evolution families have a hard time to come by.


Introduction
Evolutionary Equations as introduced in the seminal paper [11] provide a Hilbert space perspective towards numerous (both linear and non-linear) time-dependent phenomena in mathematical physics.We refer to the monographs [12,18] for a set of examples as well as further development of the theory.It is instrumental for the success of the theory of evolutionary equations that many (if not all) equations from mathematical physics can be written as a time-dependent partial differential-algebraic equation.Then, establishing the time-derivative as an m-accerive, normal operator in some weighted Hilbert space and gathering all the other unbounded operators (i.e., spatial derivative operators) in an abstract m-accretive operator A defined on some Hilbert space encoding the spatial variables, one can write evolutionary equation as an operator equation in the following form where ∂ 0 is the time-derivative, U is the unknown, F models external forces and M 0 and M 1 are linear operators in the considered space-time Hilbert space, which is a tensor product Hilbert space putting together temporal and spatial variables.Any standard solution theory for evolutionary equations of the form (1) provides conditions on the so-called material law oparators, M 0 and M 1 , so that becomes m-accretive in the space-time Hilbert space.Quickly recall that an operator T in some Hilbert space is accretive1 , if for all u ∈ dom(T ), ⟨u, Tu⟩ ≥ 0.
T is m-accretive, if T is accretive and T + λ is onto for all λ > 0 (or equivalently for some λ > 0).Looking into the different proofs under the various assumptions on the material law operators, one realises that the principal mechanism of showing m-accretivity is based on the following general well-known fact.
Theorem 1.1 (see also [5,Chapter 3,Theorem 1.43]).Let H be a Hilbert space and T : dom(T ) ⊆ H → H a densely defined and closed linear operator.Then the following conditions are equivalent: (i) T is m-accretive; (ii) T and T * are accretive.
In order to apply the last theorem showing m-accretivity of T := (∂ 0 M 0 + M 1 + A) − c, it is necessary to work out the adjoint of T , which, applying standard results, boils down to computing the adjoint of (∂ 0 M 0 + M 1 + A) in the space-time Hilbert space.Since both ∂ 0 M 0 and A are generally speaking unbounded operators, this is a non-trivial task.Our general abstract theorem provides conditions as to when for two densely defined, possibly unbounded operators S and V , we have (S +V ) * = S * +V * .This question has been addressed for instance in [8] and the references therein.In [8] criteria are provided to ensure (S +V ) * = S * +V * , which in the case of the applications in the present manuscript cannot be used since the right-hand side is (almost never) closed.Moreover, a related question is whether the sum of two unbounded operators (not necessarily adjoints) is closable and whether one can -in one way or another -obtain an expression for this closure.Questions in this range have been posed and answered in the seminal papers by [16] and [3]; we also refer to [17].In these papers also conditions for the invertibility of the operator sum are provided.The tools and results are developed for the general Banach space case for general sums, not necessarily of the sum of two adjoints of some given operators.Hence, the derived methods require conditions that are necessarily more involved compared to the present Hilbert space setting.For instance, note that closability alone for S * +V * in the present Hilbert space setting is equivalent to dom(S) ∩ dom(V ) being dense.
In any case, once a formula of the type is established, the accretivity of T (and of T * ) follow from m-accretivity of ∂ 0 M 0 +M 1 .Thus, conditions for (1) being well-posed need to address the two facts: computing the adjoint in the way sketched above needs to be possible and the problem needs to be m-accretive if A = 0.
The aim of this article is to understand the situation for the case M 0 = M 0 (m 0 ) and M 1 = M 1 (m 0 ) are multiplication operators of multiplying in the time-variable by t → M 0 (t) and t → M 1 (t), respectively.It is known that Lipschitz continuous M 0 allows for computing the adjoint as above and suitable positive definiteness conditions for M 0 together with its (a.e.existing) derivative M ′ 0 and M 1 lead to m-accretivity for the case A = 0, see [14] or [18,Chapter 16].
A particular instance of the perspective of using operator sums to understand partial differential equations has been provided (at least) as early as [16].However, the methods fail to apply in a straightforward manner as the coefficient M 0 is allowed to have a non-trivial kernel here (at least in the case of Lipschitz continuous M 0 ).We illustrate our findings in the non Lipschitz case by means of an example later on; note that this particular instance was addressed in [2].Even though we were not able to fully rectify the arguments mentioned in this reference, their major application is concerned with continuous-in-time coefficients anyway.It seems that this condition is crucial for evolution families to be applicable.Thus, in case of non-Lipschitz continuous M 0 , we establish well-posedness for a system of equations other methods (such as evolution families or evolution semigroups, see [4, Chapter VI, Section 9] and [9]) are structurally deemed to fail.
The next section is concerned with some functional analytic preliminaries, which we shall find useful in the subsequent parts.The subsequent section contains our main result concerning operator sums of m-accretive operators.The second to last section deals with applications to evolutionary equations.We summarise our findings and open problems in the conclusion section.

Preliminaries
Throughout this section, let H 0 , H 1 , H 2 be Hilbert spaces.
Lemma 2.1.Let T : dom(T ) ⊆ H 0 → H 1 be a closed linear operator and B : H 1 → H 2 bounded and linear.Then, if B is one-to-one and has closed range, BT is closed.
Proof.The closed graph theorem yields that the adstriction B : H 1 → ran(B) of B is a continuously invertible operator.Next, let (x n ) n be in dom(BT ) such that x n → x and BT x n → y in H 0 and H 2 as n → ∞ for some x ∈ H 0 and y ∈ H 2 , respectively.Then, by the continuity of ( B) −1 , we infer T By the closedness of T , we obtain x ∈ dom(T ) ⊆ dom(BT ) and T x = B −1 y.Applying B to both sides of the latter equality, we infer y = BT x as desired.
Theorem 2.2.Let T : dom(T ) ⊆ H 0 → H 1 be closed, B : H 2 → H 0 be a bounded linear operator.Then, as an identity of relations, we have Moreover, T B is densely defined if and only if B * T * is a closable operator.If, B * is one-to-one and has closed range and T is densely defined, then B * T * is closed.In particular, in this case, we have T B is densely defined and Proof.The first statement is a consequence of [18, Theorem 2.
as B * T is closed, again by Lemma 2.1.Computing adjoints on both sides, we infer where we used that T * B is closed.As a consequence of the above, we get Proposition 2.5.Let T : dom(T ) ⊆ H 0 → H 0 be densely defined and closed and B : Proof.B being a topological isomorphism, B −1 maps dense sets onto dense sets; thus Then Bx ∈ dom(T ).By assumption, we find The continuity of B * yields the assertion.

Adjoints of Sums of m-accretive Operators
This section is devoted to computing the adjoint of a sum of two (unbounded) operators T and S both densely defined and closed on a Hilbert space H.The aim is to provide conditions so that We refer to [13], where several conditions for this equality were given.In a Hilbert space H, (x n ) n in H is said to converge weakly to some x ∈ H, x n ⇀ x, if for all φ ∈ H, ⟨x n , φ ⟩ → ⟨x, φ ⟩.For a family (R ε ) ε>0 of bounded linear operators from a Hilbert spaces H 0 into H 1 , we say (R ε ) ε>0 converges in the weak operator topology to some bounded linear operator T : The main theorem, which in turn is relevant to the applications we have in mind, reads as follows: Theorem 3.1.Let T : dom(T ) ⊆ H 0 → H 1 and S : dom(S) ⊆ H 0 → H 1 be two densely defined closed operators such that dom(S) ∩ dom(T ) is dense.Moreover, we assume that there exist families Since dom(S) ∩ dom(T ) is dense, we thus infer where we have used R * ε → 1 H 0 and (K ε + Kε ) * → 0 in the weak operator topology.Since also u ε ⇀ u (use again L * ε → 1 H 1 in the weak operator topology), we infer that u ∈ dom S * + T * with Remark 3.2.If SR ε is bounded and dom(S) ∩ dom(T ) is a core for T in the above theorem, then (3) holds true.Indeed, from (2) we infer If now SR ε is bounded, the operator on the left-hand side in the above inclusion is bounded, and hence, the operator on the right-hand side is defined on H 1 , meaning that ran(L * ε ) ⊆ dom(S * ).Hence, for u ∈ dom(T ) ∩ dom(S) and v ∈ dom((S + T ) * ) we get and since dom(T ) ∩ dom(S) is a core for T , we infer that also L * ε v ∈ dom(T * ).In fact, a bounded commutator assumption for S and T as well as the condition that ε → (1 + εT ) −1 and ε → (1 + εS) −1 define uniformly bounded families of bounded linear operators on some neighbourhood of 0 is sufficient.In order to have a result readily applicable to the situation interesting for us in the following, we have opted to present a less general application of Theorem 3.1 (the most general situation provided here, is covered by Theorem 3.1 anyway).
Before we prove Theorem 3.3, we draw some elementary consequences of the commutator condition.A first consequence of this will be that S + T is densely defined.(ii) dom(S) ∩ dom(T ) is a core for T .In particular, dom(S) ∩ dom(T ) is dense in H.
Proof.For the first statement, we observe that (1 + T As a consequence, By [18, Lemma 9.3.3(a)], we deduce for all ε > 0 that The second statement is based on the observation that (1 + εS) −1 → 1 as ε → 0+ in the strong operator topology (this follows from the strong convergence on dom(S) and the uniform boundedness of the resolvents).Let now x ∈ dom(T ) and define x ε := (1 + εS) −1 x.Then, x ε → x as ε → 0+ and by part 1 of the present proposition, we deduce x ε ∈ dom(T ) and yielding that dom(T ) ∩ dom(S) is a core for T .

Applications to Evolutionary Equations
This section is devoted to apply the previous findings to operator equations in weighted, vector-valued L 2 -type spaces.The general setting can be found in [12,18].Throughout, let H be a Hilbert space and for ρ ∈ R we let endowed with the obvious norm and corresponding scalar product.We define For ρ > 0, it can be shown that ∂ 0 is m-accretive with ℜ∂ 0 = ρ.
For a bounded, strongly measurable, operator-valued function M : R → L(H), we denote by the associated multiplication operator of multiplying by M. In applications, M will be induced by scalarvalued measurable functions; that is, M ∈ L ∞ (R).We will work under the following standing hypothesis.(ii) Let M 0 , M 1 : R → L(H) be strongly measurable and uniformly bounded.
Here we have employed the custom to re-use the notation A for the (canonically) extended operator defined on L 2,ρ (R; H) with domain L 2,ρ (R; dom(A)).The aim is to study the well-posedness of nonautonomous problems of the form under suitable commutator conditions of M 0 (m 0 ) with ∂ 0 or with A.
Bounded Commutator with ∂ 0 We begin to study the case when M 0 (m 0 ) and ∂ 0 have a bounded commuator; that is, we assume there exists a strongly measurable and uniformly bounded mapping M ′ 0 : R → L(H) such that Remark 4.2.In [14] it was shown that this assumption is equivalent to the Lipschitz-continuity of M 0 .In this case, M 0 is differentiable almost everywhere and M ′ 0 is just the so-defined derivative of M 0 .
is accretive as it is the sum of two accretive operators.
In order to show that its closure is m-accretive, it suffices to show that its adjoint is also accretive.For this, we calculate its adjoint with the help of Theorem 3.1.We set Finally, is bounded, and hence, (3) holds by Remark 3.2.Thus, we can apply Theorem 3.1 and obtain Since clearly A * is accretive, it remains to prove the strict accretivity of (∂ 0 M 0 (m 0 ) + M 1 (m 0 )) * = (∂ 0 M 0 (m 0 )) * + M 1 (m 0 ) * .In order to work out the first adjoint we recall that dom(∂ 0 ) is a core for ∂ 0 M 0 (m 0 ) and hence Now, as in Lemma 4.3 one proves that (6) yields the accretivity of

Trivial Commutator with A
Here, we assume a commutator condition with A. To keep things simple, we assume that there exists d > 0 such that M 0 (t) ≥ d for almost every t ∈ R (7) and that M 0 (m 0 )A ⊆ AM 0 (m 0 ).
is m-accretive.Thus, for proving the present theorem, it suffices to apply Theorem 3.3 to T = M 0 (m 0 ) 1/2 ∂ 0 M 0 (m 0 ) 1/2 and S = A. What remains is to show the commutativity of the resolvents: where we used Hence, Theorem 3.3 is applicable and the assertion follows.
Lemma 4.9.Assume Hypothesis 4.1 together with (7).Then, for all c > 0 there exists ρ 0 > 0 such that for each ρ ≥ ρ 0 the operator Proof.For u ∈ dom(∂ 0 M 0 (m 0 ) 1/2 ) we have Choosing now ρ large enough, we infer the strict accretivity of the operator.Since its adjoint is of the form same argument shows that for ρ large enough, this operator is also accretive, and hence, the assertion follows.
Corollary 4.10.Assume Hypothesis 4.1 together with (7) and (8).Then the operator Proof.In the present situation, consider T := T + S, where and S := A. We will show that T − c is m-accretive for some c > 0. By assumption and Lemma 4.9, it is not difficult to see that T − c is accretive for some c > 0 and all large enough ρ > 0. Using the formula for the adjoint in Theorem 4.8 and taking into account the accretivtiy of T * − c, we have that T * − c is, too, accretive.Hence, 0 ∈ ρ( T ).The reformulation just before Remark 4.7 yields the assertion by multiplying T by the topological isomorphism M 0 (m 0 ) −1/2 from the left and M 0 (m 0 ) 1/2 from the right.
Remark 4.11.A similar result holds under the assumption that Next we treat a non-autonomous example of a transport equation on a graph with finitely many edges of equal length 1.The following is merely to illustrate an example, where one can have L ∞ -dependence of time-varying transport velocities in the graph.Note that well-posedness results of an autonomous version of this kind of problems are known from [6] in L 1 ; and [7] with spatially dependent velocities.The corresponding non-autonomous situation has been adressed in [1] with a weak differentiability condition on the time-dependent velocities.In the following example, we dispense with any regularity conditions on the velocity.However, instead we need rather strong commutativity properties for the velocity matrix with the matrix describing the boundary conditions.Note that time-and spatially dependent velocities so that the time-dependence is Lipschitz regular can also be dealt with within an evolutionary equations setting.Indeed, the perspective provided in [15] together with the (abstract) non-autonomous wellposedness result in [18,Theorem 16.3.1]or [14,Theorem 2.13] can be understood in this way.This particularly emphasises the interest of PDEs with L ∞ -time dependence only.

Conclusions
We provided applicable conditions for m-accretive operators so that the adjoint of the sum can be represented as the closure of the sum of adjoints of the individual operators.We applied this observation to evolutionary equations and developed well-posedness criteria for the same.The case of the operator M 0 being L ∞ -in time only and having non-trivial (possibly time-independent) kernel remains open for easily applicable conditions establishing well-posedness.Thus, to properly address the general situation of time-dependent partial differential algebraic equations appears to be a challenge for future research.