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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.

Operator theory --- Toeplitz operators --- Spectral theory (Mathematics) --- 517.984 --- Spectral theory of linear operators --- Toeplitz operators. --- Spectral theory (Mathematics). --- 517.984 Spectral theory of linear operators --- Operators, Toeplitz --- Linear operators --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Algebraic variety. --- Asymptotic analysis. --- Asymptotic expansion. --- Big O notation. --- Boundary value problem. --- Change of variables. --- Chern class. --- Codimension. --- Cohomology. --- Compact group. --- Complex manifold. --- Complex vector bundle. --- Connection form. --- Contact geometry. --- Corollary. --- Cotangent bundle. --- Curvature form. --- Diffeomorphism. --- Differentiable manifold. --- Dimensional analysis. --- Discrete spectrum. --- Eigenvalues and eigenvectors. --- Elaboration. --- Elliptic operator. --- Embedding. --- Equivalence class. --- Existential quantification. --- Exterior (topology). --- Fourier integral operator. --- Fourier transform. --- Hamiltonian vector field. --- Holomorphic function. --- Homogeneous function. --- Hypoelliptic operator. --- Integer. --- Integral curve. --- Integral transform. --- Invariant subspace. --- Lagrangian (field theory). --- Lagrangian. --- Limit point. --- Line bundle. --- Linear map. --- Mathematics. --- Metaplectic group. --- Natural number. --- Normal space. --- One-form. --- Open set. --- Operator (physics). --- Oscillatory integral. --- Parallel transport. --- Parameter. --- Parametrix. --- Periodic function. --- Polynomial. --- Projection (linear algebra). --- Projective variety. --- Pseudo-differential operator. --- Q.E.D. --- Quadratic form. --- Quantity. --- Quotient ring. --- Real number. --- Scientific notation. --- Self-adjoint. --- Smoothness. --- Spectral theorem. --- Spectral theory. --- Square root. --- Submanifold. --- Summation. --- Support (mathematics). --- Symplectic geometry. --- Symplectic group. --- Symplectic manifold. --- Symplectic vector space. --- Tangent space. --- Theorem. --- Todd class. --- Toeplitz algebra. --- Toeplitz matrix. --- Toeplitz operator. --- Trace formula. --- Transversal (geometry). --- Trigonometric functions. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Vector space. --- Volume form. --- Wave front set. --- Opérateurs pseudo-différentiels

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In the first two chapters of this book, the reader will find a complete and systematic exposition of the theory of hyperfunctions on totally real submanifolds of multidimensional complex space, in particular of hyperfunction theory in real space. The book provides precise definitions of the hypo-analytic wave-front set and of the Fourier-Bros-Iagolnitzer transform of a hyperfunction. These are used to prove a very general version of the famed Theorem of the Edge of the Wedge. The last two chapters define the hyperfunction solutions on a general (smooth) hypo-analytic manifold, of which particular examples are the real analytic manifolds and the embedded CR manifolds. The main results here are the invariance of the spaces of hyperfunction solutions and the transversal smoothness of every hyperfunction solution. From this follows the uniqueness of solutions in the Cauchy problem with initial data on a maximally real submanifold, and the fact that the support of any solution is the union of orbits of the structure.

Hyperfunctions. --- Submanifolds. --- Alexander Grothendieck. --- Analytic function. --- Analytic manifold. --- Borel transform. --- Boundary value problem. --- Bounded function. --- Bounded set (topological vector space). --- Bounded set. --- C0. --- CR manifold. --- Cauchy problem. --- Codimension. --- Coefficient. --- Cohomology. --- Compact space. --- Complex manifold. --- Complex number. --- Complex space. --- Connected space. --- Continuous function (set theory). --- Continuous function. --- Convex set. --- Convolution. --- Cotangent bundle. --- Counterexample. --- De Rham cohomology. --- Dense set. --- Differential operator. --- Disjoint union. --- Domain of a function. --- Eigenvalues and eigenvectors. --- Embedding. --- Entire function. --- Equation. --- Equivalence class. --- Equivalence relation. --- Euclidean space. --- Existential quantification. --- Exterior algebra. --- Exterior derivative. --- Fiber bundle. --- Fourier transform. --- Function space. --- Functional analysis. --- Fundamental solution. --- Harmonic function. --- Holomorphic function. --- Homomorphism. --- Hyperfunction. --- Hypersurface. --- Infimum and supremum. --- Integration by parts. --- Laplace's equation. --- Limit of a sequence. --- Linear map. --- Linear space (geometry). --- Linear subspace. --- Locally convex topological vector space. --- Mathematical induction. --- Montel space. --- Montel's theorem. --- Morphism. --- Neighbourhood (mathematics). --- Norm (mathematics). --- Open set. --- Partial derivative. --- Partial differential equation. --- Polytope. --- Presheaf (category theory). --- Pullback (category theory). --- Pullback. --- Quotient space (topology). --- Radon measure. --- Real structure. --- Riemann sphere. --- Serre duality. --- Several complex variables. --- Sheaf (mathematics). --- Sheaf cohomology. --- Singular integral. --- Sobolev space. --- Special case. --- Submanifold. --- Subsequence. --- Subset. --- Summation. --- Tangent bundle. --- Theorem. --- Topology of uniform convergence. --- Topology. --- Transitive relation. --- Transpose. --- Transversal (geometry). --- Uniform convergence. --- Uniqueness theorem. --- Vanish at infinity. --- Variable (mathematics). --- Vector bundle. --- Vector field. --- Wave front set.