Cosine families and semigroups really differ

We reveal three surprising properties of cosine families, distinguishing them from semigroups of operators: (1) A single trajectory of a cosine family is either strongly continuous or not measurable. (2) Pointwise convergence of a sequence of equibounded cosine families implies that the convergence is almost uniform for time in the entire real line; in particular, cosine families cannot be perturbed in a singular way. (3) A non-constant trajectory of a bounded cosine family does not have a limit at infinity; in particular, the rich theory of asymptotic behaviour of semigroups does not have a counterpart for cosine families. In addition, we show that equibounded cosine families that converge strongly and almost uniformly in time may fail to converge uniformly.


Introduction
The theory of cosine families is in many aspects parallel to the theory of semigroups of operators [1,20]. For example, the fundamental theorem of Sova-DaPrato-Giusti on the generation of cosine families is akin to the Hille-Yosida-Feller-Phillips-Miyadera theorem on the generation of semigroups, and both results have in fact a common source in the Henning-Neubrander representation theorem [1,4]. Similarly, the classical form of the Trotter-Kato-Neveu theorem on the convergence of semigroups carries over to cosine families with only minor changes [4,19,20,25]. However, despite similarities like those above, there are also fundamental differences between the theory of cosine families and that of semigroups. This paper will expose a number of differences between cosine families and semigroups which in our opinion are of fundamental nature.
The first of the differences that we shall reveal is embedded into a wider context of operator-valued functional equations. Functional equations satisfied by families of bounded linear operators related to Cauchy problems imply that the trajectories of the families exhibit surprising automatic continuity properties. A representative example cannot be deduced from the sole convergence of resolvents; see e.g. [2,3,5,[7][8][9]11] and references given therein.
Strikingly, examples of such an irregular behaviour have not been observed for cosine families. To discuss this in more detail, we first note that the Trotter-Kato theorem in extended form also holds for cosine families, with the notion of a regularity space playing a similar role as for semigroups [4]. We now point out that in [4] and [10] sequences of equibounded cosine families were constructed such that (i) the regularity space is a proper subset of X , (ii) the associated semigroups given by the abstract Weierstrass formula converge also outside the regularity space, and (iii) the cosine families do not converge outside the regularity space. These examples suggest that cosine families by nature cannot converge in an irregular manner like that described above with regard to semigroups. We give a precise statement of this assertion in the main result of Sect. 4. The corresponding proof will draw on Fattorini's theorem mentioned earlier. One noteworthy consequence of our result is that there is no singular perturbation theory for cosine families analogous to that for semigroups of operators.
As a final contribution in Sect. 3, we show that no non-constant trajectory of a bounded cosine family has a limit at infinity. This result, generalising [1, Proposition 3.14.6], has the consequence that the important theory of asymptotic behaviour of semigroups [15,27] does not have a cosine family analogue.
The examples presented in [4] and [10] have led to the conjecture that if equibounded cosine families converge strongly, they do so uniformly on R, and not just almost uniformly. If true, the message conveyed in this conjecture would mark another difference between semigroups and cosine families. However, as we show in the final Sect. 5, the conjecture does not hold-equibounded cosine families like equibounded semigroups may fail to converge uniformly. In particular, we give several examples of cosine families, each of which is a common strong limit of two sequences of cosine families, of which one converges uniformly and the other converges merely almost uniformly on R.

Preliminaries
We first review some of the concepts and terminology used in the paper. Let X be a Banach space. We denote by L (X ) the space of all bounded linear operators on X and by I X the identity operator on X . An L (X )-valued family is said to be a cosine family on X if The concept of a cosine family on a Banach space is closely related to the more familiar concept of a one-parameter semigroup of operators on a Banach space. A family {S(t)} t≥0 of bounded linear operators on a Banach space X is a semigroup of operators on X if 900 A. Bobrowski and W. Chojnacki J. Evol. Equ.
Our main focus throughout the paper will be on strongly continuous cosine families and semigroups on a Banach space. We recall that a family F = {F(t)} t∈T of bounded linear operators on a Banach space X indexed by T ⊂ R is strongly continuous if, for each x ∈ X , the F-trajectory (or simply trajectory) associated with x, T t → F(t)x ∈ X , is continuous in norm. We also recall that strongly continuous semigroups and cosine families on a Banach space are uniquely characterised by their respective generators. The generator A of a strongly continuous semigroup {S(t)} t≥0 on a Banach space X is defined by where D(A), the domain of A, is the set of all x ∈ X for which the derivative (2.1) exists. In turn, the generator A of a strongly continuous cosine family {C(t)} t∈R on X is defined by where D(A) is the set of all x ∈ X for which the second derivative (2.2) exists. For standard results concerning strongly continuous semigroups and cosine families and their corresponding generators, the reader is referred to, e.g. [1,18], or [20]. Some of the cosine families and semigroups that we shall consider will be bounded. We recall that a family F = {F(t)} t∈T of bounded linear operators on a Banach space We shall also make use of the concept of a strongly measurable family of operators on a Banach space. A family {F(t)} t∈T of linear bounded operators on a Banach space X indexed by a measurable set T ⊂ R is strongly measurable if, for each x ∈ X , the function t → F(t)x is Bochner measurable on T .

Measurability implies continuity
The following theorem of Fattorini, already mentioned in the Introduction, is fundamental to subsequent considerations, having a direct bearing on the issue of convergence of cosine families which is of concern in this paper. THEOREM 1 (Fattorini). A strongly measurable cosine family on a Banach space is strongly continuous on R.
As it turns out, Fattorini's theorem is an immediate consequence of the seemingly weaker, but perhaps better known theorem of Chander and Singh, also already mentioned in the Introduction, stating that a strongly measurable cosine family on a Banach Vol. 13 (2013) Cosine families and semigroups 901 space is strongly continuous on R\{0}. The link between the two theorems is provided by the following result: PROPOSITION 1. Let {C(t)} t∈R be a cosine family on a Banach space X . Let x ∈ X be such that the function t → C(t)x is continuous on (a, ∞) for some a ∈ R. Then the function t → C(t)x is continuous on R.
Proof. Fix t ∈ R arbitrarily. Choose h > 0 so that t + h > a + 1. We clearly have for each τ ∈ R. Noting that τ + h > a and τ + 2h > a whenever τ > t − 1 and exploiting the assumption, we get As C(h) is a bounded operator, we also have Since t was chosen arbitrarily, the result follows.
We remark that the above proposition has analogues for solutions of other functional equations (see e.g. the proof of [28, Thm. 2.1]), and these can in turn be used to establish suitable counterparts of Theorem 1.
Fattorini's theorem has a global flavour, in that it concerns the set of all trajectories of a cosine family. Below, we establish a local version of Fattorini's result pertaining to any individual trajectory. THEOREM 2. Let {C(t)} t∈R be a cosine family on a Banach space X . Let x ∈ X be such that the function t → C(t)x is measurable on R. Then the function t → C(t)x is continuous on R.
Proof. Let X 0 be the set of all those y ∈ X for which t → C(t)y is measurable on R. It is clear that X 0 is a linear subspace of X . Since the pointwise limit of a sequence of measurable functions is a measurable function, we see that X 0 is closed. Moreover, X 0 is invariant for each operator C(s), s ∈ R; for if s ∈ R and y ∈ X 0 , then t → C(t + s)y and t → C(t − s)y are both measurable, and consequently, t → C(t)C(s)y = (C(t + s)x + C(t − s)y)/2 is measurable, too. Now, if we consider the cosine family {C(t) |X 0 } t∈R on X 0 , where C(t) |X 0 denotes the restriction of C(t) to X 0 , then it is clear that this family is strongly measurable, and so, by Theorem 1, it is in fact strongly continuous. Since x is in X 0 , it follows that t → C(t)x is continuous, and the theorem is proved.

Convergence is regular
The Trotter-Kato theorem for cosine families, pertaining to the case where cosine families converge to a cosine family strongly and uniformly on compact time intervals, is completely analogous to the Trotter-Kato theorem for semigroups of operators (cf. [20, §8.6]; see also [19] and [25]). Based on this analogy, one might think that generally if cosine families converge, they do so in a fashion similar to that exhibited by semigroups. However, as it turns out, there is a fundamental difference in the behaviour of semigroups and cosine families with regard to convergence which reveals itself only when one considers situations more general than that captured by the Trotter-Kato theorem. Here, we expose this difference and show that the convergence of cosine families is as a rule far more regular than that of semigroups.
We start by recalling the scenario that holds for semigroups. For each n ∈ N, let A n generate a strongly continuous semigroup S n = {S n (t)} t≥0 on a Banach space X . Assume that the semigroups S n are equibounded, i.e. there is a constant M > 0 such that sup and that, for each λ > 0, the resolvents of the generators A n , converge to an operator R λ in the strong operator topology. The R λ 's then form a pseudo-resolvent-a family of operators satisfying the first resolvent equation, with a common null space and a common range. Note that, if the S n 's converge strongly (on all of X ), then the convergence of the (λ− A n ) −1 is a consequence of the representation (4.1) and the Lebesgue dominated convergence theorem. Associated with the S n 's are two subspaces of X : where the superscript "s" is reminiscent of the term "semigroup." In other words, X s point consists of vectors for which the corresponding trajectories of the semigroups converge pointwise, and X s almost-unif consists of vectors for which the corresponding trajectories of the semigroups converge almost uniformly on [0, ∞). Clearly, As it turns out, X s almost-unif coincides with the regularity space of the pseudo-resolvent {R λ } λ>0 , which is defined as the closure of the common range R of the R λ 's: Vol. 13 (2013) Cosine families and semigroups 903 with the bar denoting the set closure. Moreover, if the set X s point \X s almost-unif is nonempty, then it consists of vectors for which the corresponding trajectories of the semigroups converge almost uniformly in t ∈ (0, ∞), but not uniformly on any compact interval in [0, ∞) containing 0; see [5]. Of course, for vectors outside X s point , the corresponding trajectories of the semigroups do not converge at all. In the situation treated in the classical version of the Trotter-Kato theorem, we have R = X and then X s almost-unif = X s point = X , which is a rather special case. In general, X s almost-unif is a proper subspace of X s point , and X s point is a proper subspace of X . For example, in the space C 3 , the semigroups where (a n ) n≥1 is a sequence of non-negative numbers converging to a ≥ 0, are equibounded with bound M = 1, and the corresponding resolvents of the A n 's converge: [2,3,5,[7][8][9]11] and the references given there.
It turns out that the above picture changes when semigroups are replaced by cosine families: with the notation analogous to the one employed above and fully explained below, we have In other words, the trajectories of equibounded cosine families either converge almost uniformly in t ∈ R or do not converge at all. This dichotomy is the main point of discussion in this section. Proceeding to details, let (C n ) n∈N be a sequence of strongly continuous equibounded cosine families on X -that is, such that sup n∈N, t∈R C n (t) < ∞. Let (A n ) n∈N be the sequence of the corresponding generators. Furthermore, suppose that the resolvents of the A n 's, converge strongly to a pseudo-resolvent {R λ } λ>0 . Again, we remark that the strong convergence of the (λ − A n ) −1 's is a necessary condition for the strong convergence of the C n 's. Let and is almost uniform in t ∈ [0, ∞)}.
Then, clearly, X c almost-unif ⊂ X c point . We also have, in analogy to (4.2), as we show next. We start by pointing out that the inclusion R ⊂ X c almost-unif has already been established in [4]. To prove the reverse inclusion, for each n ∈ N, let S n = {S n (t)} t≥0 be the strongly continuous semigroup on X related to C n by the abstract Weierstrass formula Then, as is well known, the generator of S n coincides with A n (see e.g. [1, proof of Theorem 3.14.17] or [20,Theorem 8.7]). Now, if x is in X c almost-unif so that lim n→∞ C n (t)x exists for all t ∈ [0, ∞) and is almost uniform in t ∈ [0, ∞), then also lim n→∞ S n (t)x exists for all t ∈ [0, ∞) and is almost uniform in t ∈ [0, ∞), meaning that x is a member of the space X s almost-unif associated with the S n 's. But then, by (4.2), x is automatically a member of R, proving that X c almost-unif ⊂ R. With (4.6) established, we are ready for the main result of this section. THEOREM 3. For equibounded, strongly continuous cosine families C n = {C n (t)} t∈R , n ∈ N, on a Banach space X , under the assumption of strong convergence of the resolvents of the generators of the C n 's, we have Proof. We only need to show that X c point ⊂ X c almost-unif . One possible proof of this inclusion involves an argument similar to the one that leads to [5, Propositions 1 and 2]. Below, however, we shall present a much simpler proof based on (4.6).
In view of the equiboundedness of the C n 's, X c point is a closed subspace of X . For each t ∈ R, define the bounded operator C(t) : X c point → X by Vol. 13 (2013) Cosine families and semigroups 905 Invoking the equiboundedness of the C n 's again and taking into account that each C n satisfies the cosine functional equation, we deduce that for each t ∈ R, C(t) maps X c point into itself and C = {C(t)} t∈R satisfies the cosine functional equation. Thus, C is a cosine family on X c point . Being the strong limit of (C n ) n∈N , C is strongly measurable. By Theorem 1, C is in fact strongly continuous in t ∈ R. Let A be the generator of C. By the hypothesis, the definition of C, the representation (4.5), and its analogue for A, and the Lebesgue dominated convergence theorem, we have point and λ > 0; here, of course, {R λ } λ>0 is the limit pseudo-resolvent, defined on all of X , corresponding to the resolvents of the generators A n of the C n 's. are closed, it follows that X c point ⊂ X c almost-unif . The proof is now complete. The above theorem may be phrased as follows: outside the regularity space, equibounded cosine families always fail to converge strongly. In light of our earlier discussion, an analogous statement for semigroups of operators is not true. We illustrate the situation with an example which is representative of the examples mentioned in the Introduction that served as original motivation behind Theorem 3. Let ( n ) n∈N be a sequence of positive numbers converging to zero. In X = C[0, ∞], the space of continuous functions on [0, ∞) with a limit at infinity, the operators A n f = 1 2 f with domains D(A n ), composed of twice continuously differentiable functions f in X with f in X such that f (0) = n f (0) generate cosine families that are bounded in norm by 3. The resolvents of the A n 's converge strongly, and the closure of the range of the limit pseudo-resolvent is the subspace C 0 (0, ∞] of all those functions in C[0, ∞] that vanish at 0. In other words, the regularity space here is C 0 (0, ∞]. We remark in passing that the semigroups generated by the A n 's are related to elastic Brownian motions [6,23,24], and the limit semigroup on C 0 (0, ∞] is related to the minimal Brownian motion. As it turns out, the semigroups generated by the A n 's do converge strongly outside C 0 (0, ∞], with the convergence being almost uniform in t > 0 [3,5]. However, as was shown in [10], the cosine families generated by the A n 's do not converge strongly outside C 0 (0, ∞]. Theorem 3 makes it clear that this lack of convergence is not an accidental occurrence, but a rule. Here is another example illustrating the same phenomenon. For each n ∈ N, let T n (t) and B n be the restrictions of S n (t) and A n given in (4.3) and (4.4) to C 2 , respectively, i.e.

T n (t)
x y = e −a n t x e −nt y and B n x y = −a n x −ny .
Clearly, (T n ) n∈N is a sequence of equibounded semigroups generated by the B n 's, with X s almost-unif = C × {0} and X s point = C × C. The B n 's also generate equibounded cosine families, 906 A. Bobrowski and W. Chojnacki J. Evol. Equ.

C n (t)
x y = cos( √ a n t) x cos( √ nt) y , but these, unlike the T n 's, converge merely on X s almost-unif (= X c point = X c almost-unif ) which is the regularity space for the limit pseudo-resolvent corresponding to the resolvents of the B n 's. We remark that the disparity between X s almost-unif and X s point is related to the "smoothing property" of the Weierstrass formula: the semigroups derived from cosine families by means of this formula are much more "regular" than the cosine families themselves, and therefore, their trajectories typically converge for a larger sets of vectors than the trajectories of the cosine families.
We note that Theorem 3 gives information on individual trajectories as well: if, for some x ∈ X, lim n→∞ C n (t)x exists for all t ∈ R, then x ∈ X c point = X c almost-unif , and so the convergence is in fact almost uniform in t ∈ R.
One noteworthy consequence of Theorem 3 is that-in contrast to the rich theory of singular perturbations of semigroups-the theory of singular perturbations of cosine families has no objects of study: there are no sequences of cosine families that converge in an irregular way.
We conclude this section with two supplementary results. The first of these asserts that, unlike in the case of semigroups (see e.g. [27]), there is only one cosine family, a trivial cosine family, for which the strong limit lim t→∞ C(t) exists. The proof relies on an argument involving a strongly convergent sequence of cosine families, hence the inclusion of the result in this section. The second result is a local version of the first result and asserts that if a trajectory of a bounded cosine family has limits at infinity, then it is constant. An immediate consequence of this last result is that the important theory of asymptotic behaviour of semigroups [15,27] does not have a counterpart for cosine families.
We say that a cosine family C = {C(t)} t∈R on a Banach space X is trivial if C(t) = I X for all t ∈ R. If C is strongly continuous, then C is trivial if and only if the generator of C is the zero operator.

PROPOSITION 2. Let C = {C(t)} t∈R be a (non-necessarily strongly continuous) cosine family on a Banach space X such that the limit
exists for all x ∈ X . Then C is trivial and P = I X .
Proof. For each n ∈ N, let C n = {C n (t)} t∈R be the cosine family defined by C n (t) = C(nt). By assumption, (C n (t)) n∈N converges strongly to P for t > 0. Since C(−t) = C(t) for every t ∈ R, it follows that (C n (t)) n∈N converges strongly to P also for t < 0. Since, in addition, C n (0) = I X for all n ∈ N, we see that (C n (t)) n∈N converges strongly for all t ∈ R, and the limit C ∞ (t) of (C n (t)) n∈N is equal to P for t = 0 and to I X for t = 0. Since, by the Banach-Steinhaus theorem, sup n∈N C n (t) < ∞ for every t ∈ R and since each C n satisfies the cosine functional equation, we immediately conclude that C ∞ = {C ∞ (t)} t∈R satisfies the cosine functional equation. Now, if we Vol. 13 (2013) Cosine families and semigroups 907 fix s = 0 and substitute first 2s for t and next s for t in the formula 2C ∞ (t)C ∞ (s) = C ∞ (t + s) + C ∞ (t − s), then we get 2P 2 = 2P and 2P 2 = P + I X , respectively. This implies P = I X . On the other hand, letting s → ∞ in the formula 2C(t)C(s)x = C(t + s)x + C(t − s)x, we obtain 2C(t)x = 2C(t)Px = Px + Px = 2x. Since this holds for all t ∈ R and all x ∈ X , the proposition is established.
We remark that if C in the above proposition is assumed to be strongly continuous, then the equality P = I X can be established differently, in a much simpler way. Indeed, C ∞ is then strongly measurable, being the strong limit of the strongly continuous families C n , and hence, by Theorem 1, C ∞ is in fact strongly continuous. Now, P = I X is a consequence of the strong continuity of C ∞ and the fact that C ∞ (t) = P for t = 0 and C ∞ (0) = I X .

COROLLARY 1. Let C = {C(t)} t∈R be a bounded cosine family on a Banach space X , and let x ∈ X be such that the limit lim t→∞ C(t)x exists. Then C(t)x = x for all t ∈ R.
Proof. Let X 0 be composed of those elements y of X for which lim t→∞ C(t)y exists. By the boundedness of C, X 0 is a closed linear subspace of X . Exploiting the cosine functional equation, we readily deduce that each C(t), t ∈ R, leaves X 0 invariant. Since the restriction C |X 0 = {C |X 0 (t)} t∈R of C to X 0 satisfies the assumptions of Proposition 2, it follows that C(t) |X 0 = I X 0 for all t ∈ R. As x is in X 0 , we have C(t)x = x for all t ∈ R, and the corollary is proved.

Convergence may fail to be uniform
We now turn our attention to another problem related to convergence of cosine families. As shown in [10], on C 0 (0, ∞], the cosine families related to elastic Brownian motions described in the previous section converge uniformly and not merely almost uniformly on R. A similar (and more elementary) example of uniform convergence was described in [4]. Based on these results, it was conjectured that equibounded cosine families, if they converge, do so uniformly on R. If true, this would expose yet another difference between convergence of semigroups and convergence of cosine families. However, the conjecture turns out to be false, and here we show that it fails with the aid of a certain general result and two specific examples of its use, and also by means of an independent result of a slightly different nature.
A continuous function f on R with values in a Banach space X is said to be (uniformly) almost periodic if the set of its translates {T t f } t∈R is relatively compact in the metric ρ( f, g) = sup t∈R f (t) − g(t) ; the translate T t f of f by t ∈ R is given by the definition T t f (s) = f (t + s), s ∈ R. Let AP(R, X ) be the space of all X -valued almost periodic functions on R. For any f ∈ AP(R, X ) and any α ∈ R, the mean value 908 A. Bobrowski and W. Chojnacki J. Evol. Equ.
exists and defines the Fourier-Bohr coefficient of f for the Fourier exponent α, f (α). The Fourier-Bohr coefficients of f vanish for all but at most countably many Fourier exponents. The set constitutes the Bohr spectrum of f and is non-empty if f is non-zero. The function f is uniquely determined by its Fourier-Bohr coefficients, this property being implicitly meant when one refers to f via its expansion into a formal Fourier-Bohr series For a function f defined on R and λ ∈ R, we denote by f [λ] the function on R given by THEOREM 4. Let f ∈ AP(R, X ) be non-constant, where X is a Banach space, and let (λ n ) n∈N be a sequence of real numbers converging to λ ∈ R and such that λ n = λ for each n ∈ N. Then ( f [λ n ] ) n∈N converges to f [λ] almost uniformly but not uniformly on R.
Proof. The almost uniform convergence of ( f [λ n ] ) n∈N to f [λ] stems from the fact that f is uniformly continuous and the fact that the functions t → λ n t converge to t → λt almost uniformly on R. All we then need is to show that the convergence of ( f [λ n ] ) n∈N is not uniform on R.
Since f is non-zero, ( f ) is non-empty. Let (α k ) k∈ [K ] be an enumeration of ( f ), where [K ] denotes the interval of positive integers between 1 and K , with K being finite or infinite. According to Bochner's theorem [12] (see also [14,Thm. 1.24]), there exists a sequence of trigonometric polynomials which converges uniformly to f on R in the norm topology of X , as m → ∞, with r k,m being rational numbers that depend on α k and m, but not on f (α k ), and satisfy lim m→∞ r k,m = 1 for every k ∈ N. Let > 0 and let k 0 be an arbitrary member of for each t ∈ R and for each n > N 0 and each t ∈ R. Note that, λ n = 0 for all sufficiently large n (i.e. for all n > N for some N ). Indeed, when λ = 0, this is obvious, and when λ = 0, this follows from the assumption that λ n = λ for all n. Consequently, if k ∈ {1, . . . , K m }\{k 0 }, then Taking into account (5.1), we see that f (α k 0 ) ≤ 6 . Hence f (α k 0 ) = 0, by the arbitrariness of . We have thus shown that ( f ) ⊂ {0}. This implies that f is constant, a contradiction.
Let C = {C(t)} t∈R be a strongly continuous cosine family on a Banach space X . For any λ ∈ R, we denote by C [λ] the X -valued cosine family given by An element x ∈ X will be termed an almost periodic vector for C if the C-trajectory associated with x is almost periodic. We note that if (λ n ) n∈N is a sequence of real numbers converging to λ ∈ R, then (C [λ n ] ) n∈N converges to C [λ] strongly and almost uniformly in t ∈ R. Indeed, arguing as in the proof of Theorem 4, this follows from the fact that, for each x ∈ X, t → C(t)x is uniformly continuous on every compact subset of R and the fact that the functions t → λ n t converge to t → λt almost uniformly on R.
An immediate consequence of Theorem 4 and the above observation is the following.
THEOREM 5. Let C = {C(t)} t∈R be a strongly continuous cosine family on a Banach space X . Suppose that there exists an almost periodic vector for C such that the C-trajectory associated with this vector is non-constant. Let (λ n ) n∈N be a sequence of real numbers converging to λ ∈ R and such that λ n = λ for each n ∈ N. Then (C [λ n ] ) n∈N converges to C [λ] strongly and almost uniformly but not uniformly in t ∈ R.
We now present two instances of use of the above result. In the first case, we let X = C and C(t) = cos t for each t ∈ R and further observe that x = 1 is an almost periodic vector for C; then, an application of Theorem 5 leads to the following result: EXAMPLE 1. If (λ n ) n∈N is a sequence of real numbers converging to λ ∈ R and such that λ n = λ for each n ∈ N, then the sequence ({cos λ n t} t∈R ) n∈N of the scalarvalued cosine families converges to the scalar-valued cosine family {cos λt} t∈R almost uniformly but not uniformly on R.
Before discussing the other case, we introduce a class C of cosine families as follows: a cosine family C = {C(t)} t∈R on a Banach space X is in C if only if the following conditions are satisfied: (C1) there exists a sequence (C n ) n∈N , C n = {C n (t)} t∈R , of cosine families on X converging to C strongly and uniformly in t ∈ R and such that C n = C for every n ∈ N; (C2) there exists a sequence (C n ) n∈N ,C n = {C n (t)} t∈R , of cosine families on X converging to C strongly and almost uniformly but not uniformly in t ∈ R.
Vol. 13 (2013) Cosine families and semigroups 911 With this definition in hand, we can now proceed to the other instance of use of Theorem 5. Based on Theorem 5, Example 2 below reveals that the class C is nonvoid. The significance of this result is that it demonstrates that the question of whether or not the limit of a sequence of cosine families is uniform or almost uniform cannot, in general, be answered in terms of the limit cosine family alone. ξ k e k , and, for each t ∈ R, let C(t) be the linear operator on c 0 given by with ξ = (ξ k ) k∈N being an arbitrary member of c 0 . Clearly, for each n ∈ N, C n = {C n (t)} t∈R is a strongly continuous cosine family of contractions on c 0 , and also C = {C(t)} t∈R is a strongly continuous cosine family of contractions on c 0 . Given λ ∈ R and n ∈ N, we shall write C n, [λ] in lieu of (C n ) [λ] . For any n ∈ N, any t, λ ∈ R, and any ξ ∈ c 0 , we have Thus, for each λ ∈ R, the sequence (C n, [λ] ) n∈N converges to C [λ] strongly and uniformly on R.
On the other hand, the representation (5.6) implies that every ξ ∈ c 0 is an almost periodic vector for C, and moreover, the function t → C(t)ξ, t ∈ R, is non-constant whenever ξ = 0. Therefore, by Theorem 5, if, for a given λ ∈ R, (λ n ) n∈N is a sequence of real numbers converging to λ ∈ R and such that λ n = λ for each n ∈ N, then the sequence (C [λ n ] ) n∈N converges to C [λ] almost uniformly but not uniformly on R in the strong operator topology. Actually, invoking Theorem 4 directly, we can infer a bit more: for every non-zero ξ in c 0 the functions t → C [λ n ] (t)ξ, n ∈ N, converge to t → C [λ] (t)ξ almost uniformly but not uniformly in t ∈ R.
We thus see that for each λ ∈ R, C [λ] is the limit of two sequences of cosine families, each comprising cosine families different from C [λ] , such that one sequence converges uniformly, while the other converges almost uniformly but not uniformly on R; in other words, C [λ] is a member of C .
The final result of this section, namely Example 3 below, will show that for a cosine family C to belong to C it is not necessary to possess an almost periodic vector for which the C-trajectory associated with this vector is non-constant. In particular, it will become clear that membership in C can be established by means other than Theorem 5. EXAMPLE 3. We consider C 0 (0, ∞] as being equipped with the uniform norm f ∞ = sup x≥0 | f (x)|. Let C = {C(t)} t∈R be the cosine family on C 0 (0, ∞] given by for each t ≥ 0, withf being the odd extension of f on the whole of R. We claim that C is a member of C and that C has no non-zero almost periodic vector.
To prove the first assertion, let F be a real even continuous function on R with a limit at infinity. Then, in particular, F is bounded. For each ∈ R, let M F be the operator of multiplication by e F on C 0 (0, ∞]: Clearly, M F is bounded, with M F ≤ e | | F ∞ , and the inverse of M F coincides with M − F . For each ∈ R and each t ∈ R, set In view of the evenness of F, we have the following explicit expression Vol. 13 (2013) Cosine families and semigroups 913 for every t ≥ 0. We remark that it is only for the purpose of obtaining this elegant formula that we assumed F to be even-this assumption may be well dispensed with without detriment to the argument that follows. It is clear that for any ∈ R, C ( ) (t) is a linear bounded operator on C 0 (0, ∞] with C ( ) (t) ≤ e 2| | F ∞ for every t ∈ R, and C ( ) = {C ( ) (t)} t∈R is a strongly continuous cosine family on C 0 (0, ∞]. Let ( n ) n∈N be a sequence of real numbers converging to 0 and let = sup n∈N | n |.
Then, (C ( n ) ) n∈N is a sequence of equibounded cosine families on C 0 (0, ∞], with C ( n ) (t) ≤ e 2 F ∞ for each n ∈ N and each t ∈ R. Since (M − n F ) n∈N and (M n F ) n∈N tend to the identity operator on X in operator norm, we see that (C ( n ) ) n∈N converges to C in operator norm uniformly on R. Thus, to finish the proof that C is a member of C , it suffices to find a sequence of cosine families on C 0 (0, ∞] that converges to C strongly and almost uniformly but not uniformly on R. We will show that for any sequence (λ n ) n∈N of positive numbers converging to 1 and such that λ n = 1 for each n ∈ N, (C [λ n ] ) n∈N furnishes the desired sequence [we use the notation as per (5.5)].
As (C [λ n ] ) n∈N converges to C strongly and almost uniformly on R, it suffices to show that the convergence is not uniform. We shall establish the following stronger statement: for any non-zero f in C 0 (0, ∞], the functions t → C [λ n ] (t) f, n ∈ N, fail to converge uniformly to t → C(t) f . To this end, given f ∈ C 0 (0, ∞]\{0}, we first observe that for any t ≥ 0 and n ∈ N, the right-hand side of the formula if we let x = t. Thus, setting we have As sup t≥0 |f ((1 − λ n )t)| = f ∞ , our task will be complete once we show that lim n→∞ f n ∞ = 0. For any > 0, choose T > 0 so that | f (t) − f (∞)| < whenever t > T . Then, clearly, sup t>T | f n (t)| ≤ for all n. On the other hand, if t ∈ [0, T ], then |2t − (1 + λ n )t| ≤ |1 − λ n |T , and this in conjunction with the fact that f is uniformly continuous on [0, T ] implies that sup t∈[0,T ] | f n (t)| ≤ for all sufficiently large n. Thus, f n ∞ ≤ for all sufficiently large n, which finishes the proof that C is in C .
Passing to the proof of the second assertion, note that for any f ∈ C 0 (0, ∞] and any x ≥ 0. Suppose now that f ∈ C 0 (0, ∞] is an almost periodic vector for C. Then, in view of the above equalities, for each α ∈ R and each x ≥ 0. Thus, all the Fourier-Bohr coefficients of t → C(t) f vanish and consequently f = 0. The second assertion and the claim are established.
We conclude this section and the paper with two remarks. First, as it was already indicated at the beginning of this section, the operators A n specified in the paragraph following Theorem 3 generate cosine families C n on C[0, ∞] that converge to C on C 0 (0, ∞] strongly and uniformly in t ∈ R (see [10,Theorem 4.3]). Critically, the families C n do not have C 0 (0, ∞] for an invariant space, and hence, the restrictions of these families to C 0 (0, ∞] are not genuine cosine families on C 0 (0, ∞]. This explains why a sequence of cosine families different from (C n ) n∈N , namely (C ( n ) ) n∈N , was used to prove that C falls into C in the example above.
Second, we note that the same family C considered as acting on the larger Banach space BU C 0 (0, ∞) of bounded, uniformly continuous functions on [0, ∞) vanishing at 0, belongs to C , too. This can be directly deduced from Theorem 5 by observing that the sine function belongs to BU C 0 (0, ∞) and is an almost periodic vector (in fact a periodic vector, with the obvious meaning of the term) for C. While this result and that of Example 3 are similar in content, there is a difference between the arguments used for their establishment. The argument for the first result hinges on an application of Theorem 5 and proves the existence of a sequence of trajectories associated with a single vector for which the convergence is not uniform on R. In contrast, the other argument, just as the argument underpinning Example 2, shows that for no non-zero vector, the corresponding trajectories converge.