**Memoirs of the American Mathematical Society**

2004;
76 pp;
Softcover

MSC: Primary 51; 46; 37;
Secondary 35

Print ISBN: 978-0-8218-3545-6

Product Code: MEMO/171/810

List Price: $60.00

Individual Member Price: $36.00

**Electronic ISBN: 978-1-4704-0411-6
Product Code: MEMO/171/810.E**

List Price: $60.00

Individual Member Price: $36.00

# 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

#### Readership

Graduate students and research mathematicians interested in symplectic spaces and their connections to differential operators.