A note concerning a property of symplectic matrices

This note provides a counterexample to a proposition stated in [J. Differ. Equ. 261.4 (2016) 2528--2551] regarding the neighborhood of certain $4\times 4$ symplectic matrices.


Introduction
We denote by I n the n × n identity matrix, by J the standard 4 × 4 symplectic matrix, i.e. J = 0 I 2 −I 2 0 ∈ R 4×4 , and by Sp(R 4 ) = S ∈ R 4×4 : S T JS = J the corresponding symplectic group, which shall be equipped with some norm. Furthermore a matrix S ∈ Sp(R 4 ) is called elliptic if the spectrum σ(S) is contained in S 1 \ {±1}.
In section 2 we will present a continuous family of symplectic matrices contradicting the following statement: Then there exists a neighborhood U ⊂ Sp(R 4 ) of P such that a matrix S ∈ U is elliptic if and only if the following conditions hold det(S − I 4 ) > 0 and tr S < 4. ( This proposition has been used in the proof of Theorem 1.1 of [1] to obtain a spectral stability result for periodic solutions of a perturbed Kepler problem. It has not been used for the instability result contained in the same theorem.
The characteristic polynomial χ ε of P ε is given by so especially det(P ε − I 4 ) = χ ε (1) = ε 4 1 + ε 2 > 0 and we also see that the spectrum Thus there exists no neighborhood of P 0 , on which condition (2) implies ellipticity.

A Lagrangian splitting of R 4 is a decomposition
form a Lagrangian splitting of R 4 . Moreover, since P ε e 1 = e 1 + εe 2 , P ε e 2 = −εe 1 + e 2 , On the other hand in the limiting case ε = 0 the map P 0 does not admit an invariant Lagrangian splitting: Indeed let U 0 ⊕ V 0 be a splitting of R 4 into P 0 -invariant planes. We can assume that U 0 (otherwise V 0 ) contains a vector of the form u 1 = ae 1 + be 2 + e 3 + ce 4 with a, b, c ∈ R. By the invariance also u 2 = P 0 u 1 = (a + 1)e 1 + be 2 + e 3 + ce 4 ∈ U 0 . So u T 1 Ju 2 = −1 implies that the splitting is not Lagrangian.
This elaboration shows that the family (P ε ) ε∈[0,∞) contradicts also a lemma on which the proof of Proposition 1 is based: Lemma 2 (Lem. 2.5 of [1]). Let {S n } be a sequence of matrices in Sp(R 4 ) converging to S. In addition assume that for each n ≥ 0 there exists a splitting of R 4 by Lagrangian planes that are invariant under S n . Then there exists another splitting by Lagrangian planes that are invariant under S.