Characterisation of Log-Convex Decay in Non-Selfadjoint Dynamics

The short-time and global behaviour are studied for an autonomous linear evolution equation, which is defined by a generator inducing a uniformly bounded holomorphic semigroup in a Hilbert space. A general necessary and sufficient condition is introduced under which the norm of the solution is shown to be a log-convex and strictly decreasing function of time, and differentiable also at the initial time with a derivative controlled by the lower bound of the generator, which moreover is shown to be positively accretive. Injectivity of holomorphic semigroups is the main technical tool.


INTRODUCTION
The subject of this note is the global and short-time behaviour of the solutions to a linear autonomous evolution equation having a possibly non-selfadjoint generator −A.
It is assumed that A is an accretive operator with domain D(A) in a complex Hilbert space H , with norm | · | and inner product (· | ·), and that −A generates a uniformly bounded, holomorphic semigroup e −zA for z in an open sector of the form Σ δ = { z ∈ C | −δ < arg z < δ }. Then the "height" function is studied for the solution u(t) = e −tA u 0 to the following Cauchy problem, where only initial data u 0 = 0 are considered, ∂ t u + Au = 0 for t > 0, u(0) = u 0 in H .
(2) The intention is to investigate the algebraic conditions on A, which give a log-convex decay of h(t).
In a recent article on final value problems by A.-E. Christensen and the author [CJ18a], cf. also [CJ18b], it was elucidated and proved (except for one remnant) that if A is an elliptic variational operator and A is hyponormal, cf. work of Janas [Jan94], then in terms of the numerical range and the lower bound there is a "nice" decay of the height function: h(t) is strictly positive, strictly decreasing and strictly convex on the closed halfline t ≥ 0, and h(t) is differentiable at t = 0, for |u 0 | = 1 generally with h ′ (0) ≤ −m(A), though with First of all this shows how the short-time behaviour at t = 0 via h ′ (0) is specifically controlled by ν(A), the numerical range of A, and not by its spectrum σ (A); whereas the crude decay estimate h(t) ≤ Ce −tσ for t → ∞ is given by the spectral abscissa σ = inf Re σ (A) of A, say in case A * = A ≥ 0.
Secondly, the global behaviour of the height h(t) is expressed in its strict decrease and strict convexity: even if A has eigenvalues in C \ R, as may be the case, they do not induce oscillations in the size of the solution e −tA u 0 for such A-this is ruled out by strict convexity, which therefore can be seen as a stiffness in the decay of h(t).
The present paper generalises the above-mentioned results of [CJ18b,CJ18a] in three ways: First the restriction to variational generators −A is completely removed.
Secondly, the additional assumption that A is hyponormal is replaced by the weaker condition that A satisfies the following for vectors x ∈ D(A 2 ) such that |x| = 1, The third improvement is the stronger conclusion that h(t) is a log-convex function 1 . Actually condition (5) characterises the A for which h(t) is log-convex; cf. Theorem 2.6 below. Somewhat surprisingly, strict monotone decay h(t) ց 0 for t → ∞ results from (log-)convexity of h (since e −tA is uniformly bounded), hence follows whenever the generator A fulfils (5). The convexity of h then implies existence of h ′ (0) = inf h ′ < 0 and that (4) holds. The latter then shows that A is barely better than accretive (i.e. m(A) ≥ 0) in the sense that its numerical range is contained in the open right half-plane, It seems appropriate to call A a positively accretive operator, when it has the property (6). This is a milder condition on A than strict accretivity, i.e. m(A) > 0, used by Kato [Kat95]. The elliptic variational generators in [CJ18b,CJ18a] are all strictly accretive, but as described, there is no need to find a substitute assumption for this, as any A satisfying criterion (5) automatically is positively accretive.
As the terms ±|Yu| 2 cancel when the last two lines are added, (5) reduces for Here it is noteworthy that Y only appears in one term. In view of the Cauchy-Schwarz inequality, it is clear that the above is fulfilled when X and Y fit so together that the imaginary part is positive for all u ∈ H . In terms of the commutator [Y, X] = Y X − XY , one may write (10) equivalently as However, (10) and (11) are always violated for certain B ∈ B(H) when dim H ≥ 2; cf. Remark 3.4 below. So, in other words, criterion (5) is not fulfilled for every operator A in H , neither for bounded A, nor for n × n-matrices, n ≥ 2. Conversely, when A does fulfil (5), and dim H < ∞ or more generally A is a Hilbert-Schmidt operator, then A is necessarily a normal operator; cf. Remark 2.10. It is therefore envisaged that (5) can give rise to interesting examples when A is a suitable realisation of a partial differential operator.

DISCUSSION AND MAIN RESULTS
The reader is assumed familiar with semigroup theory, for which the book of Pazy [Paz83] could be a reference; the simpler Hilbert space case is exposed e.g. by Grubb [Gru09,Ch. 14]. It is briefly mentioned that there is a bijective correspondence between the C 0 -semigroups e −tA in B(H) that are uniformly bounded, i.e. e −tA ≤ M for t ≥ 0, and holomorphic in some sector Σ δ ⊂ C for δ ∈ ]0, π 2 [ , and the densely defined, closed operators A in H satisfying a resolvent estimate |λ | Log-convexity may be a new aspect in the context, so the discussion is begun with this. First it is recalled that for a strictly positive function f : R → ]0, ∞[ =: R + , log-convexity means that log f (t) is convex, that is, for all r ≤ t in R and 0 < θ < 1, As a slight extension, this also makes sense for non-negative functions f : R → [0, ∞[ . A classical exercise shows for the intermediate point s = (1 − θ )r + θt that one has θ = (s − r)/(t − r). Explicitly log-convexity therefore means for the height function that, for 0 ≤ r < s < t , 1 A fortunately inconsequential flaw in the argument given for the strict convexity in [CJ18a] is pointed out in Remark 4.1 below.
A remedy of this is provided by means of the present more general results.
The operator A is just a positive scalar if dim H = 1, so (13) is then an identity because of the functional equation of the exponential function e −tA (whereas its slightly weaker property of strict convexity is expressed via a sharp inequality, oddly enough). For dim H > 1 the inequality (13) is by no means obvious for the operator function e −tA in B(H); it is the main subject of this paper. It is noteworthy that the power function t → t θ in (13) does not require its continuous extension to t = 0, for since e −tA u 0 is holomorphic, the height function fulfils h(t) > 0, or equivalently e −tA u 0 = 0, for t ≥ 0.
This follows from the restriction to u 0 = 0 and the crucial fact that e −zA is an injection for all z ∈ Σ δ : CJ18a]). Whenever −A generates a holomorphic semigroup e −zA in B(X) for some complex Banach space X , and e −zA is holomorphic in the open sector Σ δ ⊂ C given by | arg z| < δ for some δ > 0, then the operator e −zA is injective on X for each such z.
The injectivity is for t > 0 clearly equivalent to the geometric property that two solutions e −tA v and e −tA w to the differential equation u ′ + Au = 0 cannot have any points of confluence in X when v = w. One obvious consequence of this is the backward uniqueness of u ′ + Au = 0; i.e. u(T ) = 0 implies u(0) = 0. But injectivity was seemingly first obtained in [CJ18a], cf. the elementary proof in Proposition 1 there, using unique analytic continuation. [Rau91,Cor. 4.3.9] is analogous, but is given for the Laplacian A = −∆ on Euclidean space, though for local vanishing of e t ∆ u 0 in an open set at a fixed time t > 0. (An early attempt to obtain Lemma 2.1 was made in a special case in [Sho74], but it had flaws pointed out in [CJ18a].) Injectivity of e −tA is also a crucial tool for the proof of the log-convexity in the present paper. Indeed, the fact that h(t) > 0 allows an application of the next result, that characterises the log-convex C 2 -functions as the solutions to a differential inequality: Lemma 2.2. If f is continuous [0, ∞[ → R + and C 2 for t > 0, the following properties are equivalent: ( In the affirmative case f (t) is log-convex also on the closed halfline [0, ∞[ . Remark 2.3. It is classical that a C 2 -function f is convex if and only if f ′′ ≥ 0. This positivity is fulfilled if f satisfies (I), as ( f ′ ) 2 ≥ 0 and f (t) > 0 is assumed, and it is so in qualified way, equivalent to log-convexity by Lemma 2.2. Though the lemma is not mentioned, convexity is amply elucidated in [NP06].
Proof. By the assumptions F(t) = log f (t) is defined for t ≥ 0 and C 2 for t > 0, as the Chain Rule gives Hence (I) is equivalent to F ′′ (t) ≥ 0 for t > 0, which is the criterion for the C 2 -function F to be convex for t > 0; which is a paraphase of the condition (II) for log-convexity of the positive function f (t) for t > 0. By letting r → 0 + for fixed s < t , it follows from the continuity of f (r) and of exp( To shed light on the lemma's consequences for height functions, one may conveniently use differential calculus in Banach spaces as exposed e.g. by Hörmander [Hör85, Ch. 1] or Lang [Lan72]. Note, though, that the inner product on H , despite its sesquilinearity, is differentiable on the induced real vector space tA u 0 is non-zero for all u 0 = 0 by injectivity of the semigroup, it follows from the Chain Rule for real Banach spaces, applied to the composite and hence, since The differential inequality in (I) of Lemma 2.2, is therefore equivalent to and to 2( Obviously this condition is fulfilled for every t > 0 when A satisfies condition (5) above, for u(t) = e −tA u 0 belongs to the subspace D(A n ) ⊂ D(A 2 ) for every n ≥ 2, and all u 0 ∈ H , when the semigroup is holomorphic. So in this case, it follows from Lemma 2.2 that h(t) = |e −tA u 0 | is log-convex for t ≥ 0, for the continuity of h(t) and of its derivatives given above entail that the C 2 -condition is fulfilled. Conversely, in case the height function h(t) is known to be log-convex for all u 0 = 0, then the generator −A necessarily fulfils condition (5) above. Indeed, in view of the equivalence of (19) and (21), the former of these holds by the log-convexity of h, and so does the latter. Especially it is seen by insertion of an arbitrary u 0 ∈ D(A 2 ) in (21) and commutation of A and A 2 with the semigroup that By passing to the limit for t → 0 + it follows for reasons of continuity that Hence a normalisation to x = 1 |u 0 | u 0 yields (5) for every unit vector x in D(A 2 ). Altogether this shows that (5) characterises the generators −A of uniformly bounded, holomorphic semigroups having log-convex height functions for all non-trivial initial data.
The log-convexity criterion (5) should be compared to the sufficient condition h ′′ (t) > 0 for strict convexity. The latter is seen at once from the above arguments to be equivalent to the property where in comparison to (5) the inequality is strict and a factor of 2 is absent on the left-hand side. This clearly indicates that log-convexity is stronger than strict convexity for non-constant functions: Proof. Convexity on I follows from (12) and Young's inequality for the dual exponents 1/θ and 1/(1 − θ ): In case f (r) = f (t), then the last inequality is strict, as equality holds in Young's inequality if and only if the numerators are identical (cf. [NP06,p. 14]). This yields the inequality of strict convexity in this case.
If there is a common value similarly for u ≤ s < t ; so f is strictly convex.
For completeness it is noted that for example f (t) = e t − 1 is convex, but not log-convex as (log f ) ′′ < 0. However, when f : This follows from the geometrically obvious fact that the convexity of log f survives the streching. Since f a,b clearly is not strictly convex, the last assumption of Lemma 2.4 is necessary.
When A does satisfy condition (5), so that h(t) is log-convex on [0, ∞[ for every u 0 = 0 (cf. the last part of Lemma 2.4), then h(t) is necessarily strictly decreasing: the decay estimate h(t) ≤ Ce −tη and the mere convexity statement in Lemma 2.4 show that h then satisfies the assumptions in the following self-suggesting As the strong continuity and strict decrease of h gives |e −tA u 0 | ր 1 for t → 0 + , an application of (17) yields In by commuting A with the semigroup in (17), which in the limit gives, because of the strong continuity at t = 0 and the continuity of inner products, In addition, it is seen from this that h ′ (0) that is a real number for For general u 0 ∈ H it follows from the Chain Rule that h ∈ C ∞ (R + , R).
The above discussion can now be summed up as the main result of this article: Theorem 2.6. When −A denotes a generator of a uniformly bounded, holomorphic C 0 -semigroup e −tA in a complex Hilbert space H , then the following properties are equivalent: and if u 0 ∈ D(A) with |u 0 | = 1, then Furthermore, when A has the properties (i) and (ii), then A is positively accretive, ν(A) ⊂ C + .
Returning to the case of hyponormal generators considered in [CJ18a], it is first recalled that a densely defined unbounded operator A in H , following Janas [Jan94], is said to be hyponormal if Obviously this is fulfilled if A * = A, but the hyponormal operators extend the selfadjoint operators in another direction than symmetric ones do (as these have a full operator inclusion A ⊂ A * ). Since clearly A is normal if and only if both A and A * are hyponormal, this operator class is quite general.
In case A is a hyponormal operator in H , the inclusion D(A) ⊂ D(A * ) gives at once for x ∈ D(A) that Invoking also the norm inequality from the definition of hyponormality, a similar reasoning shows for Hence, by using the Cauchy-Schwarz inequality in the above identity, one finds After a normalisation to |x| = 1, this shows that a hyponormal operator always fulfils condition (i) in Theorem 2.6, cf. (31). Therefore one has the following generalisation of [CJ18a] to the case of hyponormal non-variational generators: Relying on this remark, one can even show a similar fact for the log-convexity criterion in Theorem 2.6: Remark 2.10. When A in B(H) satisfies the log-convexity criterion (31) and n = dim H < ∞ then A is necessarily a normal operator. Indeed, in (11) the real part X is self-adjoint, so by the Spectral Theorem H has an orthonormal basis (e 1 , . . . , e n ) of eigenvectors of X . Then (11) gives 0 ≤ Im([Y, X]e j | e j ) so that the positive imaginary axis i R + contains ([Y, X]e j | e j ) for all j ∈ {1, . . . , n}. Since i R + is a convex set, any It is instructive to review condition (31) in case the accretive operator A is variational: that is, for some Hilbert space V ⊂ H algebraically, topologically and densely and some sesquilinear form a : V × V → C, which is V -bounded and V -elliptic in the sense that (with · denoting the norm in V ) for some C 0 > 0 For such operators, (A 2 u | u) = a(Au, u) and |Au| 2 = a(u, Au) holds for u ∈ D(A 2 ). So with the usual convention for the "real" part, namely a Re (v, w) = 1 2 (a(v, w) + a(w, v)) for v, w ∈ V , one has Re(A 2 u | u) + |Au| 2 = Re a(Au, u) + Rea(u, Au) = 2 Re a Re a(Au, u) .
Thus the log-convexity criterion (31) can be stated for V -elliptic variational operators in the form of a comparison of sesquilinear forms, Example 2.11. To see that variational operators need not be hyponormal, one may take H = L 2 (α, β ), with This is clearly V -bounded, and also V -elliptic: using partial integration and taking the mean of the two expressions for a(u, v), one finds Re a(u, u) = u ′ 2 0 + 1 2 |u(β )| 2 , so that Re a(u, u) ≥ C 0 u 2 1 follows for all u ∈ V and e.g. C 0 = min( 1 2 , (β − α) −2 ) by ignoring the last term and applying the Poincaré inequality (it is known that a standard proof of this, as in e.g. [Gru09,Thm. 4.29], applies to the functions in V ).
The induced A acts in the distribution space of Schwartz [Sch66] as Au = −u ′′ + u ′ , which is the advection-diffusion operator having its domain given by a mixed Dirichlet and Neumann condition, (The pure Dirichlet realisation of A = −u ′′ + u ′ has been studied at length; cf. Chapter 12 in the treatise of Embree and Trefethen [TE05], where use of pseudospectra is the main tool.) Since A * is induced by the adjoint form a * (u, v) = a(v, u), it is similarly seen that A * u = −u ′′ − u ′ , but here with the domain characterised by a mixed Dirichlet and Robin condition, As both D(A) \ D(A * ) = / 0 and D(A * ) \ D(A) = / 0, it follows from (36) that neither A nor A * is hyponormal. This is part of the motivation for the introduction of the general condition (31) in this paper.

ACCRETIVE SQUARES
The considerations in [CJ18b,CJ18a] also dealt with variational operators A that, instead of being hyponormal, have accretive squares, The discussion in Section 2 also extends to such operators without the assumption that A is variational, albeit only strict convexity is obtained for h(t). Indeed, when m(A 2 ) ≥ 0 holds, then it is seen from (18) and Cauchy-Schwarz' inequality that Of course the mere convexity of h for t > 0 is implied by the above inequality h ′′ (t) ≥ 0, so As h is continuous on [0, ∞[ , this extends to 0 ≤ r < s < t , so h is convex on [0, ∞[ . Hence Lemma 2.5 also here applies to h, yielding its strict decrease. The arguments below Lemma 2.5 then apply verbatim, which leads to differentiability at t = 0 etc. of h(t) (skipping the reference to Lemma 2.4 here). Moreover, this also yields that A is positively accretive. However, it remains to prove h(t) strictly convex on [0, ∞[ when A 2 is accretive (because of the factor 2 on the left-hand side of (31), this condition is hardly implied by m(A 2 ) ≥ 0). If e.g. ν(A 2 ) ⊂ C + , clearly the first inequality in (47) is strict, i.e. h ′′ (t) > 0 for t > 0. Thus h is strictly convex for such A.
However, by inspection of the formula above, h ′′ (t) = 0 is seen to imply that both Re(A 2 u | u) ≥ 0 and (|Au||u|) 2 − (Re(Au | u)) 2 ≥ 0 must hold with equality in the first numerator. But then the inequalities hold with equality. As Cauchy-Schwarz' inequality is an identity only for proportional vectors, there is some λ = µ + iω , µ, ω ∈ R, such that Au(t) = λ u(t). Insertion of this into the equation h ′′ (t) = 0 yields which reduces to µ 2 = 0.
(51) Hence λ = i ω is an eigenvalue of A, as u(t) = e −tA u 0 = 0 in view of the restriction to u 0 = 0 and injectivity of e −tA ; cf. Lemma 2.1. But it was seen above that A is positively accretive, so it cannot have any eigenvalues on i R. Consequently h ′′ (t) > 0 holds for all t > 0, so h(t) is strictly convex for t > 0.
To extend the strict convexity to the closed halfline where t ≥ 0, one may conveniently take recourse to the slope function S(r,t) = (h(t) − h(r))/(t − r). Because of the Mean Value Theorem and the strict increase of h ′ , this satisfies S(r, s) < S(s,t) whenever 0 < r < s < t ; which is a classical criterion for strict convexity of h on ]0, ∞[ . But this sharp inequality extends to the case r = 0, for by introducing some r ′ such that r = 0 < r ′ < s < t , one finds from the convexity of h on [0, ∞[ obtained after (48) that Indeed, the first of these inequalities is valid since the slope function S(s,t) is monotone increasing in both arguments separately for every convex function on an interval. Hence h is strictly convex on [0, ∞[ . Altogether this proves a result analogous to Theorem 2.6, but not quite as strong as this, for operators A with accretive squares: Proposition 3.1. If −A denotes a generator of a uniformly bounded, holomorphic semigroup e −tA in a complex Hilbert space H and A has an accretive square, that is Here h ′′ (t) > 0 was mentioned explicitly, as not all strictly convex functions fulfil this (cf. t 4 ), whereas in Theorem 2.6 this property was straightforward from the differential inequality characterising log-convexity.
The last fact in Proposition 3.1 that A is positively accretive can post festum be much sharpened: for an accretive operator A to have an accretive square, cf. (46), it is necessary that A has semiangle δ ≤ π 4 , that is, | Im z| ≤ Re z for every z ∈ ν(A). This was shown already by Showalter [Sho74, Lem. 3], who gave the main lines in the proof of the following Lemma 3.2. If A is an operator in H so that A, A 2 are accretive, then | arg z| ≤ π/4 for all z ∈ ν(A).
As motivation for stating Lemma 3.2 and giving a concise proof (without the assumption, made in [Sho74], that −A should generate a C 0 -semigroup), it should be mentioned that, contrary to the claim in [Sho74], having semiangle δ ≤ π/4 does not suffice for A 2 to be accretive.
As the fourth order term λ 2 1 s 4 cancels on both sides, the term of highest degree is s 3 λ 1 (λ 1 − λ 2 ) on the right-hand side. After division by s 3 and passage to the limit s → ∞, one therefore arrives at the false statement "0 ≤ λ 1 (λ 1 − λ 2 )". Consequently the operator A from Example 3.3 does not fulfil the logconvexity criterion in Theorem 2.6 for any of the considered values of the parameters. Especially this is so for the matrix given at the end of Example 3.3.

FINAL REMARKS
Remark 4.1. When ν(A 2 ) ⊂ C + as in Section 3, it is also illuminating to observe the possibility to depart from the cleaner expression of the derivatives of the squared height h(t) 2 = |e −tA u 0 | 2 : Here e −tA is injective, and A is so as ν(A) ⊂ C + , whence u 0 = 0 yields (h 2 ) ′′ > 0. Similarly (h 2 ) ′′ > 0 can be seen from (39) to hold if A is hyponormal. That is, h 2 is in both cases strictly convex for t > 0. But as √ · is concave (not convex), strict convexity of h 2 is not simply carried over to h. As the task is to prove h ′ strictly increasing, the formula h ′ = (h 2 ) ′ /(2 √ h 2 ) looks convincing as there is strict decrease of the denominator while the numerator is increasing -but the formula does not lead to the desired conclusion because (h 2 ) ′ < 0.
This small point was overlooked in [CJ18a,Prop. 4], yet the statement there is nevertheless correct. Indeed, [CJ18a,Prop. 4] is generalised to non-variational A having accretive squares in Proposition 3.1, and to non-variational hyponormal generators A in Corollary 2.8. A further generalisation to generators A satisfying the log-convexity criterion (31) is provided by Theorem 2.6.
Remark 4.2. For matrices A in B(C n ) the dynamical properties of (2) have been studied for decades, and e.g. Perko [Per01, Ch. 1] gave a concise treatment with many explicit formulas for the exponential matrix e −tA and the resulting solution u(t). However, most systems have eigenvalues that are complicated or even impossible to write down (n ≥ 5), and this led Moler and Van Loan to review the possibilities in 1978 in "Nineteen dubious ways to calculate the exponential of a matrix", with an update in 2003 [MVL03].
The present results are closer in spirit to more recent work, a glimpse of which is given here, following the inspiring exposition of Embree and Trefethen [TE05, Ch. 14]. A major subject of interest has been the behaviour of the operator norm E(t) = e −tA , which has the advantage of being independent of any initial data u 0 , thereby letting the influence of especially non-normal matrices shine though. At t = 0 it is a main result that E ′ (0) = −m(A), (61) which also shows the role of the numerical range ν(A) of A. For t → ∞ it is known that log E(t) → −σ , where again σ = inf Re σ (A) denotes the spectral abscissa of A; so the long-term behaviour is controlled by the spectrum of A. For the transition phase there is the pseudospectral estimate sup t≥0 E(t) ≥ α ε (−A)/ε , supplied with estimates from below of sup 0≤s≤t E(s) that permit an exploration of the time t 0 at which sup t≥0 E(t) is attained.
However, when u 0 is reintroduced, the inequality |e −tA u 0 | ≤ E(t)|u 0 | is a crude estimate (a worst-case scenario), which does not suffice to settle whether h(t) = |e −tA u 0 | has properties like strict decrease or log-convexity as in Theorem 2.6. Moreover, (61) is often highly misleading for the short- This sharp contrast between the properties of the solutions to u ′ + Au = 0, u(0) = u 0 , and those of e −tA also motivates the study of the height function h(t) = |e −tA u 0 | in the present paper.
Remark 4.3. If the generator −A of the uniformly bounded holomorphic semigroup is dissipative, i.e. A is accretive, then e −tA is a classical contraction semigroup; cf. [Gru09, Cor. 14.11]. That is, e −tA ≤ 1 holds for the operator norm for t ≥ 0, whence h(t) ≤ e −tA · |u 0 | ≤ |u 0 |-i.e. an estimate by a constant. If m(A) > 0 it is also classical that −(A − εI) for 0 < ε < m(A) gives the contractions e −t(A−εI) = e tε e −tA , so the sharper estimate |e −tA u 0 | ≤ e −tε |u 0 | holds for t ≥ 0 and any ε ∈ ]0, m(A)[ , hence also for ε = m(A). But this exponential decay is just a crude estimate that requires m(A) > 0. For comparison it is observed that if A just satisfies the log-convexity criterion (31), so that m(A) = 0 is possible and Theorem 2.6 applies, the log-convex and strictly decreasing behaviour of the height function |e −tA u 0 | constitutes a rather more precise dynamical property of the evolution problem u ′ + Au = 0, u(0) = u 0 .