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The Spectral Theory of Toeplitz Operators. (AM-99), Volume 99
L. Boutet de Monvel, Victor Guillemin
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
Beschreibung
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.
Kundenbewertungen
Cotangent bundle, Periodic function, Operator (physics), Support (mathematics), Equivalence class, One-form, Homogeneous function, Complex manifold, Limit point, Todd class, Parametrix, Polynomial, Big O notation, Change of variables, Toeplitz operator, Symplectic geometry, Dimensional analysis, Natural number, Exterior (topology), Invariant subspace, Integral curve, Toeplitz matrix, Volume form, Hamiltonian vector field, Eigenvalues and eigenvectors, Spectral theorem, Holomorphic function, Projective variety, Symplectic manifold, Integer, Elliptic operator, Symplectic vector space, Symplectic group, Curvature form, Fourier transform, Contact geometry, Existential quantification, Codimension, Self-adjoint, Boundary value problem, Trace formula, Complex vector bundle, Diffeomorphism, Open set, Fourier integral operator, Vector space, Trigonometric functions, Theorem, Cohomology, Connection form, Scientific notation, Summation, Vector bundle, Lagrangian (field theory), Asymptotic expansion, Line bundle, Differentiable manifold, Spectral theory, Asymptotic analysis, Tangent space, Projection (linear algebra), Toeplitz algebra, Quadratic form, Quotient ring, Hypoelliptic operator, Algebraic variety, Integral transform, Linear map, Metaplectic group, Pseudo-differential operator