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Written by an author who was at the forefront of developments in multi-variable spectral theory during the seventies and the eighties, this guide sets out to describe in detail the spectral mapping theorem in one, several and many variables. The basic algebraic systems – semigroups, rings and linear algebras – are summarised, and then topological-algebraic systems, including Banach algebras, to set up the basic language of algebra and analysis. Spectral Mapping Theorems is written in an easy-to-read and engaging manner and will be useful for both the beginner and expert. It will be of great importance to researchers and postgraduates studying spectral theory.

Banach algebras. --- Mappings (Mathematics) --- Spectral theory (Mathematics) --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Maps (Mathematics) --- Functions --- Functions, Continuous --- Topology --- Algebras, Banach --- Banach rings --- Metric rings --- Normed rings --- Banach spaces --- Topological algebras --- Operator theory. --- Operator Theory.

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This book focuses on the constructive and practical aspects of spectral methods. It rigorously examines the most important qualities as well as drawbacks of spectral methods in the context of numerical methods devoted to solve non-standard eigenvalue problems. In addition, the book also considers some nonlinear singularly perturbed boundary value problems along with eigenproblems obtained by their linearization around constant solutions. The book is mathematical, poising problems in their proper function spaces, but its emphasis is on algorithms and practical difficulties. The range of applications is quite large. High order eigenvalue problems are frequently beset with numerical ill conditioning problems. The book describes a wide variety of successful modifications to standard algorithms that greatly mitigate these problems. In addition, the book makes heavy use of the concept of pseudospectrum, which is highly relevant to understanding when disaster is imminent in solving eigenvalue problems. It also envisions two classes of applications, the stability of some elastic structures and the hydrodynamic stability of some parallel shear flows. This book is an ideal reference text for professionals (researchers) in applied mathematics, computational physics and engineering. It will be very useful to numerically sophisticated engineers, physicists and chemists. The book can also be used as a textbook in review courses such as numerical analysis, computational methods in various engineering branches or physics and computational methods in analysis.

Eigenvalues. --- Spectral theory (Mathematics) --- Mathematics. --- Math --- Science --- Functional analysis --- Hilbert space --- Measure theory --- Transformations (Mathematics) --- Matrices --- Computer science --- Numerical analysis. --- Differential Equations. --- Computer science. --- Computational Mathematics and Numerical Analysis. --- Numerical Analysis. --- Ordinary Differential Equations. --- Applications of Mathematics. --- Computational Science and Engineering. --- Numerical and Computational Physics, Simulation. --- Informatics --- 517.91 Differential equations --- Differential equations --- Mathematical analysis --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Mathematics --- Computer mathematics. --- Differential equations. --- Applied mathematics. --- Engineering mathematics. --- Physics. --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Engineering --- Engineering analysis

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The theory of D-modules deals with the algebraic aspects of differential equations. These are particularly interesting on homogeneous manifolds, since the infinitesimal action of a Lie algebra consists of differential operators. Hence, it is possible to attach geometric invariants, like the support and the characteristic variety, to representations of Lie groups. By considering D-modules on flag varieties, one obtains a simple classification of all irreducible admissible representations of reductive Lie groups. On the other hand, it is natural to study the representations realized by functions on pseudo-Riemannian symmetric spaces, i.e., spherical representations. The problem is then to describe the spherical representations among all irreducible ones, and to compute their multiplicities. This is the goal of this work, achieved fairly completely at least for the discrete series representations of reductive symmetric spaces. The book provides a general introduction to the theory of D-modules on flag varieties, and it describes spherical D-modules in terms of a cohomological formula. Using microlocalization of representations, the author derives a criterion for irreducibility. The relation between multiplicities and singularities is also discussed at length.Originally published in 1990.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Differentiable manifolds. --- D-modules. --- Representations of groups. --- Lie groups. --- Groups, Lie --- Lie algebras --- Symmetric spaces --- Topological groups --- Group representation (Mathematics) --- Groups, Representation theory of --- Group theory --- Modules (Algebra) --- Differential manifolds --- Manifolds (Mathematics) --- Affine space. --- Algebraic cycle. --- Algebraic element. --- Analytic function. --- Annihilator (ring theory). --- Automorphism. --- Banach space. --- Base change. --- Big O notation. --- Bijection. --- Bilinear form. --- Borel subgroup. --- Cartan subalgebra. --- Cofibration. --- Cohomology. --- Commutative diagram. --- Commutative property. --- Commutator subgroup. --- Complexification (Lie group). --- Conjugacy class. --- Coproduct. --- Coset. --- Cotangent space. --- D-module. --- Derived category. --- Diagram (category theory). --- Differential operator. --- Dimension (vector space). --- Direct image functor. --- Discrete series representation. --- Disk (mathematics). --- Dot product. --- Double coset. --- Eigenfunction. --- Eigenvalues and eigenvectors. --- Endomorphism. --- Euler operator. --- Existential quantification. --- Fibration. --- Function space. --- Functor. --- G-module. --- Gelfand pair. --- Generic point. --- Hilbert space. --- Holomorphic function. --- Homomorphism. --- Hyperfunction. --- Ideal (ring theory). --- Infinitesimal character. --- Inner automorphism. --- Invertible sheaf. --- Irreducibility (mathematics). --- Irreducible representation. --- Levi decomposition. --- Lie algebra. --- Line bundle. --- Linear algebraic group. --- Linear space (geometry). --- Manifold. --- Maximal compact subgroup. --- Maximal torus. --- Metric space. --- Module (mathematics). --- Moment map. --- Morphism. --- Noetherian ring. --- Open set. --- Presheaf (category theory). --- Principal series representation. --- Projective line. --- Projective object. --- Projective space. --- Projective variety. --- Reductive group. --- Riemannian geometry. --- Riemann–Hilbert correspondence. --- Right inverse. --- Ring (mathematics). --- Root system. --- Satake diagram. --- Sheaf (mathematics). --- Sheaf of modules. --- Special case. --- Sphere. --- Square-integrable function. --- Sub"ient. --- Subalgebra. --- Subcategory. --- Subgroup. --- Summation. --- Surjective function. --- Symmetric space. --- Symplectic geometry. --- Tensor product. --- Theorem. --- Triangular matrix. --- Vector bundle. --- Volume form. --- Weyl group.

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Many of the operators one meets in several complex variables, such as the famous Lewy operator, are not locally solvable. Nevertheless, such an operator L can be thoroughly studied if one can find a suitable relative parametrix--an operator K such that LK is essentially the orthogonal projection onto the range of L. The analysis is by far most decisive if one is able to work in the real analytic, as opposed to the smooth, setting. With this motivation, the author develops an analytic calculus for the Heisenberg group. Features include: simple, explicit formulae for products and adjoints; simple representation-theoretic conditions, analogous to ellipticity, for finding parametrices in the calculus; invariance under analytic contact transformations; regularity with respect to non-isotropic Sobolev and Lipschitz spaces; and preservation of local analyticity. The calculus is suitable for doing analysis on real analytic strictly pseudoconvex CR manifolds. In this context, the main new application is a proof that the Szego projection preserves local analyticity, even in the three-dimensional setting. Relative analytic parametrices are also constructed for the adjoint of the tangential Cauchy-Riemann operator.Originally published in 1990.The Princeton Legacy Library uses the latest print-on-demand technology to again make available previously out-of-print books from the distinguished backlist of Princeton University Press. These editions preserve the original texts of these important books while presenting them in durable paperback and hardcover editions. The goal of the Princeton Legacy Library is to vastly increase access to the rich scholarly heritage found in the thousands of books published by Princeton University Press since its founding in 1905.

Pseudodifferential operators. --- Functions of several complex variables. --- Solvable groups. --- Analytic function. --- Analytic set. --- Associative property. --- Asymptotic expansion. --- Atkinson's theorem. --- Banach space. --- Bilinear map. --- Boundary value problem. --- Bounded function. --- Bounded operator. --- Bump function. --- C space. --- CR manifold. --- Cauchy problem. --- Cauchy's integral formula. --- Cauchy–Schwarz inequality. --- Cayley transform. --- Characteristic function (probability theory). --- Characterization (mathematics). --- Coefficient. --- Cokernel. --- Combinatorics. --- Complex conjugate. --- Complex number. --- Complexification (Lie group). --- Contact geometry. --- Convolution. --- Darboux's theorem (analysis). --- Darboux's theorem. --- Diagram (category theory). --- Diffeomorphism. --- Difference "ient. --- Differential operator. --- Dimension (vector space). --- Dirac delta function. --- Eigenvalues and eigenvectors. --- Elliptic operator. --- Equation. --- Existential quantification. --- Explicit formulae (L-function). --- Factorial. --- Fourier inversion theorem. --- Fourier series. --- Fourier transform. --- Fundamental solution. --- Heisenberg group. --- Hermitian adjoint. --- Hilbert space. --- Hodge theory. --- Hypoelliptic operator. --- Hölder's inequality. --- Implicit function theorem. --- Integral transform. --- Invertible matrix. --- Leibniz integral rule. --- Lie algebra. --- Mathematical induction. --- Mathematical proof. --- Mean value theorem. --- Multinomial theorem. --- Neighbourhood (mathematics). --- Neumann series. --- Nilpotent group. --- Orthogonal transformation. --- Orthonormal basis. --- Oscillatory integral. --- Paley–Wiener theorem. --- Parametrix. --- Parity (mathematics). --- Partial differential equation. --- Partition of unity. --- Plancherel theorem. --- Polynomial. --- Power function. --- Power series. --- Product rule. --- Property B. --- Pseudo-differential operator. --- Pullback (category theory). --- Quadratic form. --- Regularity theorem. --- Riesz transform. --- Schwartz space. --- Scientific notation. --- Self-adjoint operator. --- Self-adjoint. --- Sesquilinear form. --- Several complex variables. --- Singular integral. --- Special case. --- Summation. --- Support (mathematics). --- Symmetrization. --- Theorem. --- Topology. --- Triangle inequality. --- Unbounded operator. --- Union (set theory). --- Unitary transformation. --- Variable (mathematics).