Memoirs of the American Mathematical Society 2004; 76 pp; softcover Volume: 171 ISBN10: 0821835459 ISBN13: 9780821835456 List Price: US$57 Individual Members: US$34.20 Institutional Members: US$45.60 Order Code: MEMO/171/810
 Complex symplectic spaces, defined earlier by the authors in their AMS Monograph, are nontrivial generalizations of the real symplectic spaces of classical analytical dynamics. These spaces can also be viewed as nondegenerate indefinite inner product spaces, although the authors here follow the lesser known exposition within complex symplectic algebra and geometry, as is appropriate for their prior development of boundary value theory. In the case of finite dimensional complex symplectic spaces it was shown that the corresponding symplectic algebra is important for the description and classification of all selfadjoint boundary value problems for (linear) ordinary differential equations on a real interval. In later AMS Memoirs infinite dimensional complex symplectic spaces were introduced for the analysis of multiinterval systems and elliptic partial differential operators. In this current Memoir the authors present a selfcontained, systematic investigation of general complex symplectic spaces, and their Lagrangian subspaces, regardless of the finite or infinite dimensionalitystarting with axiomatic definitions and leading towards general GlazmanKreinNaimark (GKN) theorems. In particular, the appropriate relevant topologies on such a symplectic space \(\mathsf{S}\) are compared and contrasted, demonstrating that \(\mathsf{S}\) is a locally convex linear topological space in terms of the symplectic weak topology. Also the symplectic invariants are defined (as cardinal numbers) characterizing \(\mathsf{S}\), in terms of suitable Hilbert structures on \(\mathsf{S}\). The penultimate section is devoted to a review of the applications of symplectic algebra to the motivating of boundary value problems for ordinary and partial differential operators. The final section, the Aftermath, is a review and summary of the relevant literature on the theory and application of complex symplectic spaces. The Memoir is completed by symbol and subject indexes. Readership Graduate students and research mathematicians interested in symplectic spaces and their connections to differential operators. Table of Contents  Introduction: motivation and organization of results
 Complex symplectic spaces: fundamental concepts and definitions
 Symplectic weak topology
 Algebraic and arithmetic invariants: Hilbert structures
 Applications to the theory of symmetric linear operators
 Aftermath
 Acknowledgements
 Bibliography
 Index
