Zero-two law for cosine families

For $\left(C(t)\right)_{t \geq 0}$ being a strongly continuous cosine family on a Banach space, we show that the estimate $\limsup_{t\to 0^{+}}\|C(t) - I\|<2$ implies that $C(t)$ converges to $I$ in the operator norm. This implication has become known as the zero-two law. We further prove that the stronger assumption of $\sup_{t\geq0}\|C(t)-I\|<2$ yields that $C(t)=I$ for all $t\geq0$. Additionally, we derive alternative proofs for similar results for $C_{0}$-semigroups.


Introduction
Let (T (t)) t≥0 denote a strongly continuous semigroup on the Banach space X with infinitesimal generator A. It is well-known that the inequality This has become known as zero-one law for semigroups. Surprisingly, the same law holds for general semigroups on semi-normed algebras, i.e., (1.1) implies (1.2), see e.g. [5]. For a nice overview and related results, we refer the reader to [4].
In this paper we study the zero-two law for strongly continuous cosine families on a Banach space, i.e. whether (1.3) lim sup t→0 + C(t) − I < 2 implies that lim sup Theorem 1.1. Let (C(t)) t≥0 be a strongly continuous cosine family on the Banach space X. Then implies that lim t→0 + C(t) − I = 0.
By taking X = ℓ 2 and it is easy to see that this result is optimal. Whether one can get rid of the assumption that the cosine family is strongly continuous remains open. The zero-one law for semigroups and the zero-two law for cosine families tells something about the behaviour near t = 0. Instead of studying the behaviour around zero, we could study the behaviour on the whole time axis. A result dating back to the sixties is the following; for a semigroup the assumption implies that T (t) = I for all t ≥ 0, see e.g. Wallen [13] and Hirschfeld [8]. This result seems not to be well-known among researchers working in the area of strongly continuous semigroup. The corresponding result for cosine families, i.e., (1.6) sup t∈R C(t) − I < 2 implies that C(t) = I is hardly studied at all. We prove this result for strongly continuous cosine families on Banach spaces. This result is strongly motivated by the recent work of A. Bobrowski and W. Chojnacki. In [3,Theorem 4], they showed that if r < 1 2 , where (1.7) r = sup t≥0 C(t) − cos(at)I , then C(t) = cos(at)I for all t ≥ 0. They used this to conclude that scalar cosine families are isolated points within the space of bounded strongly continuous cosine families acting on a fixed Banach space, equipped with the supremum norm. Hence we show that for a = 0 the r can be chosen be 2, provided C is strongly continuous. We remark that by using the proof idea in [1, Theorem 1.1 in Three Line Proofs] the implication sup t∈R C(t) − I < r implies that C(t) = I holds for r < 3 2 for any cosine family. While this paper was being revised, we heard that A. Bobrowski, W. Chojnacki and A. Gregosiewicz showed that for a = 0 the implication (1.8) sup t∈R C(t) − cos(at)I < r implies that C(t) = cos(at)I holds for general cosine families with r = 8

√
3 . This constant is optimal, as can be directly seen by choosing C(t) = cos(3at)I. In [11] we wrongly claimed that r = 2 was the optimal constant.
The lay-out of this paper is as follows. In Section 2 we prove the zero-two law for strongly continuous cosine families, i.e., Theorem 1.1 is proved. In Section 3 we prove the implication (1.6). Furthermore, we give elementary, alternative proofs for strongly continuous semigroups. Throughout the paper, we use standard notation, such as σ(A) and ρ(A) for the spectrum and resolvent set of the operator A, respectively. Furthermore, for λ ∈ ρ(A), R(λ, A) denotes (λI − A) −1 .

The zero-two law at the origin
In this section we prove that for the strongly continuous cosine family C on the Banach space X Theorem 1.1 holds; i.e., However, before we do so, we first recall the definition of a strongly continuous cosine family. For more information we refer to [2] or [7].
Definition 2.1. A family C = (C(t)) t∈R of bounded linear operators on X is called a cosine family when the following two conditions hold (1) C(0) = I, and (2) For all t, s ∈ R there holds It is defined to be strongly continuous, when for all x ∈ X and all t ∈ R we have Similar as for strongly continuous semigroups we can define the infinitesimal generator.
Definition 2.2. Let C be a strongly continuous cosine family, then the infinitesimal generator A is defined as t 2 with its domain consisting of those x ∈ X for which this limit exists.
This infinitesimal generator is a closed, densely defined operator. For the proof of Theorem 1.1, the following well-known estimates are needed. For a proof we refer to Lemma 5.5 and 5.6 in [7] . Lemma 2.3. Let C be a strongly continuous cosine family with generator A. Then, there exists ω ≥ 0 and M ≥ 1 such that Furthermore, for Re λ > ω we have λ 2 ∈ ρ(A) and Hence the above lemma shows that the spectrum of A must lie within the parabola {s ∈ C | s = λ 2 with Re λ = ω}. To study the spectral properties of the points within this parabola, we can use the following lemma. Lemma 2.4. Let C be a strongly continuous cosine family on the Banach space X and let A be its generator. Then, for λ ∈ C and s ∈ R there holds (1) S(λ, s) defined by is a linear and bounded operator on X and its norm satisfies The bounded operators S(λ, s) and C(s)x − cosh(λs)I commute.
With the use of the above lemma we can show that the spectrum of A is contained in the intersection of a ball and a parabola, provided (1.4) holds, i.e., lim sup t→0 + C(t) − I < 2.
Lemma 2.5. Let C be a strongly continuous cosine family on the Banach space X with generator A. Assume that there exists c > 0 such that Then, there exists M c , r c > 0 and φ c ∈ (0, π 2 ) such that Proof. First, we note that by (2.8) we have that there exists a t 0 > 0 such that C(t) − I < c for all t ∈ [0, t 0 ), and by symmetry, for all t ∈ (−t 0 , t 0 ). Since c < 2, we conclude that 1 2 . This implies that −1 ∈ ρ(C(t)). By standard spectral theory it follows that the open ball centered at −1 with radius R(−1, C(t)) −1 is included in ρ(C(t)). Therefore, , and by the analyticity of the resolvent, we have for µ ∈ B 2−c 2 (−1) and t ∈ (−t 0 , t 0 ) that Since cosh(iπ) = −1, by continuity there exists ball in the complex plane with center iπ which is mapped under the cosh inside the ball around −1. That is, there exists ar > 0 such that (2.13) cosh(Br(iπ)) ⊂ B 1−c 2 (−1).
Combining the results from Lemmas 2.3 and 2.5 enable us to prove Theorem 1.1. As for semigroups we can prove a slightly more general result. Theorem 2.6 (Zero-two law for cosine families). Let C be a strongly continuous cosine family on the Banach space X. Denote by A its infinitesimal generator. Then the following are equivalent (1) The following inequality holds (2) The following equality holds (3) A is a bounded operator.
Proof. Trivially the second item implies the first one. If the assertion in part 3 holds, then the corresponding cosine family is given by C(t) = ∞ n=0 A n (−1) n t 2n (2n)! . From this the property in item 2 is easy to show. Hence it remains to show that item 1 implies item 3.

Similar laws on R
In this previous sections we showed that uniform estimates in a neighbourhood of zero implied additional properties. In this section we study estimates which hold on R or (0, ∞). We show that by applying a scaling trick, the results can be obtained from the already proved laws. The main theorem of this section is the following.
Theorem 3.1. The following assertions hold (1) For a semigroup T we have that (1.5) implies that T (t) = I for all t ≥ 0.
(2) If the strongly continuous cosine family C on the Banach space X satisfies then C(t) = I for all t.
Proof. Since the construction of the proof in the two items is very similar, we concentrate on the second one.
For the Banach space X we define ℓ 2 (N; X) as With the norm this is a Banach space. On this extended Banach space we define C ext (t), t ∈ R as Hence it is a diagonal operator with scaled versions of C on the diagonal. To prove that C ext is strongly continuous, we take an arbitrary x ∈ ℓ 2 (N; X) and t ∈ R. Furthermore, we choose an ε > 0 and a z = (z n ), with only finitely many z n unequal to zero, such that x − z ≤ ε. By the construction of ℓ 2 (N; X) this is always possible. Now we find since by (3.1), the cosine family C ext is bounded by 3. Let N be such that z n = 0 for n > N . Then since C is a strongly continuous cosine family. Combining this with (3.4) we find that lim sup Since ε is arbitrarily, we conclude that C ext is a strongly continuous cosine family on ℓ 2 (N; X). Now we estimate the distance from this cosine family to the identity on ℓ 2 (N; X) for t ∈ (0, 1]. where we have used (3.1). In particular, this implies that By Theorem 2.6, we conclude that the infinitesimal generator of C ext is bounded.
Since C ext (t) is a diagonal operator, it is easy to see that its infinitesimal generator A ext is diagonal as well. Furthermore, the n'th diagonal element equals nA. Since n runs to infinity, A ext can only be bounded when A = 0. This immediately implies that C(t) = I for all t.
From the above proof it is clear that if Theorem 2.6 would hold for non-strongly continuous cosine families, then the strong continuity assumption can be removed from item 2 in the above theorem as well.
As follows from the first item, for semigroups no continuity assumption was needed. As mentioned in the introduction, this can also be proved using operator algebraic result going back to Wallen [13]. In the following subsection, we present some alternative proofs, showing that they can be asked as an exercise in a first course on semigroup theory.

3.1.
Elementary proofs for semigroups. We now give some elementary proofs of the following result.
Theorem 3.2. Let T be a strongly continuous semigroup on the Banach space X, and let A denote its infinitesimal generator. If Proof of Theorem 3.2, frequency domain. Since the C 0 -semigroup is bounded by (3.5), (0, ∞) ⊂ ρ(A) and for λ > 0 we have that where we used (3.5). Thus λ(λI − A) −1 − I ≤ r Since r < 1, we know that I + λ(λI − A) −1 − I is boundedly invertible, and the norm of this inverse is less or equal to (1 − r) −1 . Hence So for all λ > 0 we have that I − λ −1 A ≤ 1 1−r . This can only hold if A = 0. Proof of Theorem 3.2, time domain. In general it holds that For t > 0 let B t denote the bounded operator x → B t x := t 0 T (s)xds. Since for all x ∈ X, x − T (s)x ds ≤ r x , and r < 1, it follows that t −1 B t is boundedly invertible for all t > 0 and .