Let \(\mathrm{Sp}(2n)\) denote the group of linear transformations of \(\mathbb{R}^{2n}\) that preserve the standard symplectic form \(\omega\). In its Lie algebra, the subset \[ {sp}^{+}(2n)=\{ X \in {sp}(2n) \ | \ \omega(X,\cdot) \ \text{is positive definite} \} \ \] is a convex cone and paths whose tangent vector lies in \({sp}^{+}(2n)\) play a key role in the study of linear Hamiltonian systems and their stability. Of particular interest for the theory of linear Hamiltonian systems are strongly stable systems. These are linear Hamiltonian systems whose solutions remain bounded for all times such that this property survives small perturbations of the system. Such systems can be studied in terms of Krein-theory. In particular, a linear symplectic map is the initial value of the matrizant of a periodic strongly stable linear Hamiltonian system if and only if all its (complex) eigenvalues lie on \(S^{1}\setminus \{\pm 1\}\) and the Krein-form \(\kappa(v,w):=\langle iJv,w \rangle\) is definite on the (complex) eigenspaces, where \(\langle \cdot, \cdot\rangle\) denotes the standard Hermitian product on \(\mathbb{C}^{2n}\) and \(J\) is the canonical complex structure on \(\mathbb{R}^{2n}\). The set of all symplectic maps such that the Krein-form is positive definite on the eigenspaces of all eigenvalues with non-negative imaginary part is an open subset of \(\mathrm{Sp}(2n)\) called the positively elliptic region \(\mathrm{Sp}^{+}_{\mathrm{ell}}(2n)\).
The main result of this paper is that \(\mathrm{Sp}^{+}_{\mathrm{ell}}(2n)\) is globally hyperbolic, in the sense that there exists an embedded hypersurface \(\Sigma\) in \(\mathrm{Sp}^{+}_{\mathrm{ell}}(2n)\) such that any inextensible curve tangent to the cone \(\mathfrak{sp}^{+}(2n)\) intersects \(\Sigma\) in a unique point.