This book presents original research results on pseudodifferential operators.
C*-algebras generated by pseudodifferential operators with piecewise smooth symbols on a smooth manifold are considered. For each algebra, all the equivalence classes of irreducible representations are listed; as a consequence, a criterion for a pseudodifferential operator to be Fredholm is stated, the topology on the spectrum is described, and a solving series is constructed.
Pseudodifferential operators on manifolds with edges are introduced, their properties are considered in details, and an algebra generated by the operators is studied.
An introductory chapter includes all necessary preliminaries from the theory of pseudodifferential operators and C*-algebras.
Author(s): Boris Plamenevskii, Oleg Sarafanov
Series: Pseudo-Differential Operators, 15
Publisher: Birkhäuser
Year: 2023
Language: English
Pages: 248
Contents
Introduction
1 Preliminaries
1.1 Pseudodifferential Operators
1.1.1 Amplitudes
1.1.2 Pseudodifferential Operators
1.1.3 The Kernel of a Pseudodifferential Operator
1.1.4 Smoothing Operators
1.1.5 Properly Supported Pseudodifferential Operators
1.1.6 Pseudodifferential Operators in Generalized Function Spaces
1.1.7 Symbols
1.1.8 Asymptotic Expansions in the Classes Sμ(Ω)
1.1.9 The Symbol of Proper Pseudodifferential Operator
1.1.10 Symbolic Calculus of Pseudodifferential Operators
1.1.11 Change of Variables in Pseudodifferential Operators
1.1.12 Classes bμ(Rn)
1.1.13 The Boundedness of DO in L2(Rn)
1.1.14 DO in Sobolev Spaces
1.1.15 Elliptic Pseudodifferential Operators
1.2 Meromorphic Pseudodifferential Operators
1.2.1 Integral Transforms on a Sphere
1.2.2 Canonical Meromorphic Pseudodifferential Operators
1.2.3 The Kernel of a Canonical Pseudodifferential Operator
1.2.4 Operations on Canonical Meromorphic Pseudodifferential Operators
1.2.5 General Meromorphic Pseudodifferential Operators
1.2.6 Change of Variables in Meromorphic Pseudodifferential Operators
1.3 C*-algebras
1.3.1 C*-algebras and Their Morphisms
1.3.2 Representations of C*-algebras
1.3.3 Spectrum of C*-algebra
1.3.4 Criteria for an Element of an Algebra to Be Invertible or to Be Fredholm
1.3.5 Continuous Field of C*-algebras
1.3.6 A Sufficient Triviality Condition for the Fields of Elementary Algebras
1.3.7 Solvable Algebras
1.3.8 Maximal Radical Series
1.3.9 The Localization Principle
2 C*-algebras of Pseudodifferential Operators on Smooth Manifolds with Discontinuities in Symbols Along a Submanifold
2.1 Algebras Generated by Pseudodifferential Operators with Smooth Symbols
2.2 Algebras of Pseudodifferential Operators with Isolated Singularities in Symbols
2.2.1 Algebras A and S
2.2.2 Proof of the Inclusion C0(R)K L2(Sn-1)S
2.2.3 Auxiliary Results
2.2.4 The Spectrum of Algebra S
2.2.5 The Spectrum of Algebra A
2.3 Algebras of Pseudodifferential Operators with Discontinuities in Symbols Along a Submanifold
2.3.1 The Statement of Basic Theorems
2.3.2 Algebras L(θ): Irreducibility
2.3.3 Localization in the Algebra L (θ)
2.3.4 The Spectrum of Algebra L(θ)
2.3.5 The Spectrum of the Algebra of Pseudodifferential Operators with Symbols Discontinuous Along a Submanifold
3 Algebra of Pseudodifferential Operators with Piecewise Smooth Symbols on a Smooth Manifold
3.1 Algebra A and Its Irreducible Representations
3.1.1 Stratification of Manifold M. Algebra A
3.1.2 The Irreducible Representations of the Algebra A (Formulation of a Theorem)
3.1.3 Proof of Theorem 3.1.4
3.2 The Spectral Topology of Algebra A
3.2.1 Description of the Jacobson Topology (Formulation of the Theorem)
3.2.2 Proof of Theorem 3.2.1
3.3 Solving Series
3.3.1 Construction of a Solving Series. Formulation of the Theorem
3.3.2 Proof of Theorem 3.3.1
Verification of Formulas (3.3.23)
4 Pseudodifferential Operators on Manifolds with Smooth Closed Edges
4.1 Pseudodifferential Operators in Rmn
4.1.1 Amplitudes
4.1.2 Pseudodifferential Operators
4.1.3 Symbols
4.1.4 Composition of do: Adjoint Operator
4.1.5 Conditions for do to Belong to Classes -∞0 and -∞
4.1.6 Elliptic do
4.2 Operators on Manifolds with Wedges
4.2.1 Admissible Diffeomorphisms of Subsets of Rmn
4.2.2 Change of Variables in do
4.2.3 Pseudodifferential Operators on a Wedge
4.2.4 W-manifolds
4.2.5 do on w-manifold
4.3 Pseudodifferential Operators in Weighted Spaces
4.3.1 Boundedness of Proper do of Non-positive Order in the Spaces L2, τ
4.3.2 Pseudodifferential Operators in the Spaces Hsτ
4.3.3 Pseudodifferential Operators on Spaces with Weighted Norms on w-Manifolds
5 C*-Algebra of Pseudodifferential Operators on Manifold with Edges
5.1 Classes Ψμ
5.2 C*-Algebra Generated by Proper do. Local Algebras
5.3 Algebras L (θ, Rn) and L (0, Rn)
5.3.1 A Special Representation of Generators of L (0, Rn )
5.3.2 Coincidence of Algebras L(θ, Rn) and L0(θ)
5.4 Localization in L (θ, K)
5.5 Localization in L(0, K)
5.6 Invariant Description of Local Algebras
5.7 The Spectrum of C*-Algebra of Pseudodifferential Operators on Manifold with Edges
Bibliographical Sketch
References
