Complex Symplectic Geometry with Applications to Ordinary Differential Operators

Complex symplectic spaces, and their Lagrangian subspaces, are defined in accord with motivations from Lagrangian classical dynamics and from linear ordinary differential operators; and then their basic algebraic properties are established. After these purely algebraic developments, an Appendix presents a related new result on the theory of self-adjoint operators in Hilbert spaces, and this provides an important application of the principal theorems. 1. Fundamental definitions for complex symplectic spaces, and three motivating illustrations Complex symplectic spaces, as defined below, are non-trivial generalizations of the real symplectic spaces of Lagrangian classical dynamics [AM], [MA]. Further, these complex spaces provide important algebraic structures clarifying the theory of boundary value problems of linear ordinary differential equations, and the theory of the associated self-adjoint linear operators on Hilbert spaces [AG], [DS], [NA]. These fundamental concepts are introduced in connection with three examples or motivating discussions in this first introductory section, with further technical details and applications presented in the Appendix at the end of this paper. The new algebraic results are given in the second and main section of this paper, which developes the principal theorems of the algebra of finite dimensional complex symplectic spaces and their Lagrangian subspaces. A preliminary treatment of these subjects, with full attention to the theory of self-adjoint operators, can be found in the earlier monograph of these authors [EM]. Definition 1. A complex symplectic space S is a complex linear space, with a prescribed symplectic form [:], namely a sesquilinear form (i) u, v → [u : v], S × S → C, so [c1u + c2v : w] = c1[u : w] + c2[v : w], (1.1) which is skew-Hermitian, (ii) [u : v] = −[v : u], so [u : c1v + c2w] = c̄1[u : v] + c̄2[u : w] Received by the editors August 19, 1997. 1991 Mathematics Subject Classification. Primary 34B05, 34L05; Secondary 47B25, 58F05.