# Infinite Dimensional Complex Symplectic Spaces

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*W. N. Everitt; L. Markus*

Complex symplectic spaces, defined earlier by the
authors in their AMS Monograph, are non-trivial generalizations of the
real symplectic spaces of classical analytical dynamics. These spaces
can also be viewed as non-degenerate 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 self-adjoint 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 multi-interval systems and elliptic
partial differential operators.

In this current Memoir the authors present a
self-contained, systematic investigation of general complex symplectic
spaces, and their Lagrangian subspaces, regardless of the finite or
infinite dimensionality—starting with axiomatic definitions and
leading towards general Glazman-Krein-Naimark (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.

#### Table of Contents

# Table of Contents

## Infinite Dimensional Complex Symplectic Spaces

- Contents vii8 free
- Infinite dimensional complex symplectic spaces 112 free
- 1. Introduction: motivation and organization of results 112
- 2. Complex symplectic spaces: fundamental concepts and definitions 314
- 3. Symplectic weak topology 1829
- 4. Algebraic and arithmetic invariants: Hilbert structures 3849
- 5. Applications to the theory of symmetric linear operators 5869
- 6. Aftermath 6374
- 7. Acknowledgements 6980
- Bibliography 7182
- Index 7384 free