The spectral theory of Toeplitz operators
Author(s)
Bibliographic Information
The spectral theory of Toeplitz operators
（Annals of mathematics studies, no. 99）
Princeton University Press , University of Tokyo Press, 1981
- : hbk
- : pbk
Available at 65 libraries
Note
Includes bibliographies
Description and Table of Contents
- Volume
-
: pbk ISBN 9780691082790
Description
The theory of Toeplitz operators has come to resemble more and more in recent years the classical theory of pseudodifferential operators. For instance, Toeplitz operators possess a symbolic calculus analogous to the usual symbolic calculus, and by symbolic means one can construct parametrices for Toeplitz operators and create new Toeplitz operators out of old ones by functional operations. If P is a self-adjoint pseudodifferential operator on a compact manifold with an elliptic symbol that is of order greater than zero, then it has a discrete spectrum. Also, it is well known that the asymptotic behavior of its eigenvalues is closely related to the behavior of the bicharacteristic flow generated by its symbol. It is natural to ask if similar results are true for Toeplitz operators. In the course of answering this question, the authors explore in depth the analogies between Toeplitz operators and pseudodifferential operators and show that both can be viewed as the "quantized" objects associated with functions on compact contact manifolds.
Table of Contents
*Frontmatter, pg. i*TABLE OF CONTENTS, pg. v* 1. Introduction, pg. 1* 2. GENERALIZED TOEPLITZ OPERATORS, pg. 11* 3. FOURIER INTEGRAL OPERATORS OF HERMITE TYPE, pg. 21* 4. THE METAPLECTIC REPRESENTATION, pg. 27* 5. METALINEAR AND METAPLECTIC STRUCTURES ON MANIFOLDS, pg. 35* 6. ISOTROPIC SUBSPACES OF SYMPLECTIC VECTOR SPACES, pg. 39* 7. THE COMPOSITION THEOREM, pg. 44* 8. THE PROOF OF THEOREM 7.5, pg. 51* 9. PULL-BACKS, PUSH-FORWARDS AND EXTERIOR TENSOR PRODUCTS, pg. 66* 10. THE TRANSPORT EQUATION, pg. 77* 11. SYMBOLIC PROPERTIES OF TOEPLITZ OPERATORS, pg. 80* 12. THE TRACE FORMULA, pg. 90* 13. SPECTRAL PROPERTIES OF TOEPLITZ OPERATORS, pg. 99* 14. THE HILBERT POLYNOMIAL, pg. 110* 15. SOME CONCLUDING REMARKS, pg. 123*BIBLIOGRAPHY, pg. 128*APPENDIX: QUANTIZED CONTACT STRUCTURES, pg. 131*Backmatter, pg. 161
- Volume
-
: hbk ISBN 9780691082844
Description
The theory of Toeplitz operators has come to resemble more and more in recent years the classical theory of pseudodifferential operators. For instance, Toeplitz operators possess a symbolic calculus analogous to the usual symbolic calculus, and by symbolic means one can construct parametrices for Toeplitz operators and create new Toeplitz operators out of old ones by functional operations.
If P is a self-adjoint pseudodifferential operator on a compact manifold with an elliptic symbol that is of order greater than zero, then it has a discrete spectrum. Also, it is well known that the asymptotic behavior of its eigenvalues is closely related to the behavior of the bicharacteristic flow generated by its symbol.
It is natural to ask if similar results are true for Toeplitz operators. In the course of answering this question, the authors explore in depth the analogies between Toeplitz operators and pseudodifferential operators and show that both can be viewed as the "quantized" objects associated with functions on compact contact manifolds.
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