THE ANALOGY OF CAMERON-JOHNSON’S

THEOREM FOR THE LINEAR ℂ-ANALYTIC

EQUATIONS IN THE HILBERTIAN SPACE.

D.N.Cheban

Abstract. The well-known Cameron-Johnson’s theorem affirms that the equation x′ = A(t)x (∗) with recurrent (almost periodic by Bhor) matrix may be reduced by the Lyapunov-Perron transformation to the equation y′ = B(t)y with skewsymmetric matrix B(t), if all solutions of the equation (*) and all its limit equations are bounded on the whole real axis. The generalization of this result on the linear ℂ-analytic equations in the Hilbertian space is given.

The well-known Cameron-Johnson’s theorem [1,2] affirms that the equation

x′ = A(t)x  (1)

with recurrent (almost periodic by Bhor) matrix may be reduced by the Lyapunov-Perron transformation to the equation y′ = B(t)y with skew-symmetric matrix B(t), if all solutions of the equation (1) and all its limit equations are bounded on the whole real axis. The generalization of this result on the linear ℂ-analytic equations in the Hilbertian space is our main scope in the article.

Let ℝ = (−∞, +∞), ℂ^{m} − m−dimensional complex Euclidean space, G be a domain from ℂ^{m}, ℍ be a real or complex Hilbertian space with scalar product < ·, · > and with norm | · |^{2} =< ·, · >. Denote by ℍ_{w} a space ℍ equipped with weak topology, and by [ℍ] ( respectively [ℍ_{w}] ) the family of all linear continuous operators acting into ℍ ( resp. ℍ_{w}) and equipped with operational norm ( with weak topology ). Assume that H(G, [ℍ])) ( resp. H(G, [ℍ_{w}]), H(G, ℂ^{m})) is the family of all holomorphic functions h : G → [ℍ] ( resp. h : G → [ℍ_{w}], h : G → ℂ^{m} ) equipped with compact-open topology and consider the system

\{ \begin{matrix}x′ & = A(z)x \\ z′ & = Φ(z) \end{matrix}  (z ∈ G),  (2)

where Φ ∈ H(G, ℂ^{m}) and A ∈ H(G, [ℍ]). We suppose that the second equation of system (2) generates the dynamical system (G, ℝ, σ) on G. Denote by U(t, z) the Cauchy’s operator of equation

x′ = A(zt)x (z ∈ G),  (3)

Typeset by AMS-TEX

1991 Mathematics Subject Classification. 34C35;34C27;34K15;34K20;58F27;34G10.

Key words and phrases. nonautonomous linear dynamical systems, almost periodic system, Cameron-Johnson’s theorem, bounded solutions, reducibility.

where zt = σ(t, z). It follows from general property of solutions of differential equations (see, for example, [3-5]) that the family of operators {U(t, z)|t ∈ ℝ, z ∈ G} satisfies the following conditions:

1. U(0, z) = I (∀z ∈ G), where I is a unit operator on ℍ ;

2. U(t + τ, z) = U(t, zτ)U(τ, z) (∀t, τ ∈ ℝ and z ∈ G );

3. the mapping U : ℝ × G → [ℍ] (U : (t, z) ↦ U(t, z)) is continuous and for every t ∈ ℝ the mapping U(t, ·) : G → [ℍ] is holomorphic.

The following assertion takes place.

Theorem 1. Suppose that there exists a positive canstant C such that

∥U(t, z)∥ ≤ C  (4)

for all t ∈ ℝ and z ∈ G, then there exists P ∈ H(G, [ℍ]) such that:

a. P - is biholomorphic, i.e. the operator P (z) is invertible for all z ∈ G and the mapping P^{−1} : G → [ℍ] (P^{−1} : z ↦ P^{−1}(z)) is holomorphic;

b. P_{∗}(z) = P (z) for all z ∈ G, where P_{∗}(z) is an operator adjoint for P (z); c. the operator P (z) is positive definite for all z ∈ G; d. C^{−1}|x| ≤ |P (z)x| ≤ C|x| for all z ∈ G and x ∈ ℍ; e. the change of variables x = P (zt)y takes equation (3) into

y′ = B(zt)y  (5)

with skew-Hermitian operator B ∈ H(G, [ℍ]), i.e. B_{∗}(z) = −B(z) for all z ∈ G.

Proof. Let S^{t} : H(G, [ℍ]) → H(G, [ℍ]) ( resp. H(G, [ℍ_{w}]) → H(G, [ℍ_{w}])) be a mapping defined by equality (S^{t}f)(z) = U_{∗}(t, z)f(zt)U(t, z) for all z ∈ G and f ∈ H(G, [ℍ]) (resp. f ∈ H(G, [ℍ_{w}])). It is easy to verify that the family of mappings {S^{t}|t ∈ ℝ} satisies the following conditions:

4. S^{0} = I, where I is a unit mapping on H(G, [ℍ]) ( resp. H(G, [ℍ_{w}]));

5. S^{t}S^{τ} = S^{t+τ} for all t, τ ∈ ℝ;

6. the mapping S^{t} ( t ∈ ℝ) is linear and continuous on H(G, [ℍ]) ( resp. H(G, [ℍ_{w}])).

It follows from 4.-6. that the family of mappings {S^{t}|t ∈ ℝ} is a commutative group of linear continuous operators on H(G, [ℍ]) ( resp. H(G, [ℍ_{w}])).

Denote by K = {A ∈ [ℍ]| ∥A∥ ≤ C^{2}} and we note that the set K is weak closed. According to Tihonov’s theorem the set K is weakly compact, because every bounded and closed set in ℍ is weakly compact. It follows from Cauchy’s integral formula (see, for example, [6,p.339]) and theorem of Ascoli [6] that the set Γ = {γ ∈ H(G, [ℍ]| ∥γ(z)∥ ≤ C^{2}} is compact in the space H(G, [ℍ_{w}]). We suppose that V = \overline{conv}\{S^{t}Q|t ∈ ℝ\}, where Q(z) = I(∀z ∈ G), I is a unit operator in ℍ and \overline{conv} is a weak closure of the convex hulle of \{S^{t}Q|t ∈ ℝ\}. We note that the set V is invariant, i.e. S^{t}V ⊆ V for all t ∈ ℝ, because the family of operators {S^{t}} is linear and continuous in H(G, [ℍ_{w}]). It is easy to see that

< S^{t}Q(z)x, x >=< U_{∗}(t, z)U(t, z)x, x >= |U(t, z)x|^{2}  (6)

for all z ∈ G, t ∈ ℝ and x ∈ ℍ. It follows from (4) and (6) that

C^{−2}|x|^{2} ≤< S^{t}Q(z)x, x >≤ C^{2}|x|^{2}  (7)

The analogy of Cameron-Johnson’s theoerem … 3

and, consequently,

C^{−2}|x|^{2} ≤< R(z)x, x >≤ C^{2}|x|^{2}  (8)

for all R ∈ V, z ∈ G and x ∈ ℍ. In addition, the equality R_{∗}(z) = R(z) takes place for all z ∈ G and R ∈ V. According to equality (8) we have ∥R(z)∥ ≤ C^{2} for all z ∈ G and, consequently, V ⊆ Γ. According to the theorem of Markov-Kakutany [7] the group {S^{t}|t ∈ ℝ} admits in V at last one fixe-point, i.e. there exists R ∈ V such that U_{∗}(t, z)R(zt)U(t, z) = R(z) for all z ∈ G and t ∈ ℝ. We note that the operator R(z) is self-adjoint and according to equality (8) it is positive definite. It follows from Theorem 12.33 [8] (see also [9,p.65]) that there exists a unique revertible selfadjoint and positive definite operator M(z) such that M^{2}(z) = R(z) for all z ∈ G and according to inequality (8) we have

C^{−1}|x| ≤ |M(z)x| ≤ C|x|

for all z ∈ G and x ∈ ℍ. Finally, according to equality

R^{α}(z) = −\frac{1}{2πi} ∮_{L} λ^{α}(R(z) − λI)^{−1}dλ

(α = ±\frac{1}{2}), where L is a simple contour enclosing the spectrum of operator R(z), we have that the mapping M : G → [ℍ] is biholomorhic.

We note that M(zt)U(t, z)M^{−1}(z) = M^{−1}(zt)U_{∗}^{−1}(t, z)M(z) because

U_{∗}(t, z)M^{2}(zt)U(t, z) = M^{2}(z)

for all t ∈ ℝ and z ∈ G. We put V(t, z) = M(zt)U(t, z)M^{−1}(z) and we note that the family of operators {V(t, z)|t ∈ ℝ, z ∈ G} satisfies the condition 1.-3. and in addition:

8.V_{∗}(t, z) = V^{−1}(t, z) for all t ∈ ℝ and z ∈ G;

9. U(t, z)P (z) = P (zt)V(t, z) for all t ∈ ℝ and z ∈ G, where P (z) = M^{−1}(z). Denote by

B(z) = \frac{d}{dt}V(t, z)|_{t=0} = [Ṁ(z) + M(z)A(z)]P (z),  (9)

where Ṁ(z) = \frac{d}{dt}M(zt)|_{t=0} = \frac{dM(z)}{dz}Φ(z), then B ∈ H(G, [ℍ]) and

\frac{dV(t, z)}{dt} = B(zt)V(t, z)  (10)

for all t ∈ ℝ and z ∈ G and, consequently, the operator V(t, z) is a Cauchy’s operator of equation (5). On the other hand, from condition 8. we have

\{ \begin{matrix}V′_{∗}^{−1}(t, z) & = −B_{∗}(zt)V_{∗}^{−1}(t, z) \\ V_{∗}^{−1}(0, z) & = I \end{matrix}  (11)

for all t ∈ ℝ and z ∈ G. It follows from the equalities (10),(11) and condition 8. that B_{∗}(z) = −B(z) for all z ∈ G. Finally, to finish the proof of theorem it is sufficient to remark that the change of variables x = P (zt)y takes equation (3) into equation (5). The theorem ic proved.

Remark. a. If the point z ∈ G is stationary ( ω-periodic, quasiperiodic, almost periodic etc ), then according to [10] the operator-function P (zt) will be also stationary (ω-periodic, quasiperiodic, almost periodic etc ).

b. The Theorem 1 takes place for the system of difference equations

\{ \begin{matrix}x(k + 1) & = A(zk)x(k) \\ z(k + 1) & = Φ(zk) \end{matrix} (k ∈ ℤ),

where A ∈ H(G, [ℍ]) and Φ ∈ H(G, ℂ^{m}) and also for the system of differential and difference equations with multidimensional time.

References

[1] Cameron R.H., Almost periodic properties of bounded solutions of linear differential equations with almost periodic coefficients., J.Math.Phys. 1936, v.15,pp.73-81.

[2] Johnson R.A., On a Floquet theory for almost periodic, two-dimensional linear systems., J. Dif. Equat. 1980, v.37, pp.184-205.

[3] Coddington E.A., Levinsion N., Theory of ordinary differential equations, Springer, Berlin, 1963.

[4] Demidovich B.P., Lecture on mathematical theory of stability, Nauka, Moscow, 1967 (in Russian).

[5] Daletskii Yu.L., Krein M.G., Stability of solutions of differential equations in Banach space, Nauka, Moscow, 1970 (English transl., Translations of Mathematical Monographs, vol. 43, Amer.Math.Soc.,Providence, R.I. 1974 ).

[6] Schwartz L., Analyse mathematique, v.2, Hermann, Paris, 1967.

[7] Bourbaki N., Espaces vectoriels topologiques, Hermann, Paris, 1955.

[8] Rudin W., Functional Analysis, McGraw-Hill Book Company, New York - London - Montreal - Singapore, 1973.

[9] Halmos P.R., A Hilbert space problem book, D.Van Nostrand Company, INC, Princeton-New Jersey-Toronto-London, 1967.

[10] Scherbakov B.A., Poisson stable motions of dynamical systems and solutions of differential equations, Shtiintsa, Kishinev, 1985 (in Russian).

D.N.Cheban State University of Moldova Faculty of Mathematics and Informatics 60, A.Mateevich str. Chişinău, MD-2009 MOLDOVA

E-mail address: cheban@usm.md