Operator Theory

Nonclassical Problems

Sergei G. Pyatkov

• Publisher: De Gruyter
• Year: 2013
• Print ISBN: 9783110628678
• eBook ISBN: 9783110900163
• Edition: 1
• Topic: Mathematics
• Subtopic: Functional Analysis
• Index: Mathematics

This monograph describes mathematical methods applicable to studying nonclassical problems of mathematical physics. The emphasis of the book is on applications of the interpolar theory of Banach spaces to the theory of linear operators to be expotentially dichotomous, to some continuity properties of linear operators in Hilbert scales, to the Riesz basis property of eigenelements and associated elements of linear pencils and the correspondending elliptic problems with indefinite weight functions, and to studying nonclassical boundary value problems for first order operator-differential equations.

Table of contents

• Chapter 1. Indefinite inner product spaces. Linear operators. Interpolation
• 1. Indefinite inner product spaces
• 1.1. Definitions
• 1.2. Krein spaces
• 1.3. The Gram operator. W-spaces
• 1.4. J-orthogonal complements. Projective completeness
• 1.5. J-orthonormalized systems
• 2. The basic classes of operators in Krein spaces
• 2.1. J-dissipative operators
• 2.2. J-selfadjoint operators
• 3. Interpolation of Banach and Hilbert spaces and applications
• 3.1. Preliminaries
• 3.2. Continuity of some functional in a Hilbert scale
• 3.3. Separation of the spectrum of an unbounded operator
• 3.4. Interpolation properties of bases
• 4. The existence of maximal semidefinite invariant subspaces for J-dissipative operators
• 5. First order equations. Decomposition of a solution
• 5.1. Function spaces
• 5.2. The Cauchy problem
• 5.3. Auxiliary definitions. Some properties of imaginary powers of operators
• 5.4. Solvability of the Cauchy problem in the original Banach space
• 5.5. Adjoint problems
• 5.6. Arbitrary operators. Phase spaces
• 5.7. Remarks and examples
• Chapter 2. Spectral theory for linear selfadjoint pencils
• 1. Examples
• 1.1. Selfadjoint pencils
• 1.2. Elliptic eigenvalue problems with indefinite weight function
• 2. Basic assumptions. The structure of root subspaces
• 3. The Riesz basis property. Invariant subspaces
• 3.1. Basis property
• 3.2. Invariant subspaces. Some applications
• 4. Sufficient conditions
• Chapter 3. Elliptic eigenvalue problems with an indefinite weight function
• 1. Auxiliary function spaces. Interpolation
• 1.1. Definitions
• 1.2. Interpolation of weighted Sobolev spaces
• 1.3. Inequalities of the Hardy type
• 2. Preliminaries. Basic assumptions
• 2.1. Variational statement
• 2.2. Elliptic problems
• 3. Basisness theorems
• 3.1. The general case
• 3.2. The one-dimensional case
• 4. Examples and counterexamples
• Chapter 4. Operator-differential equations
• 1. Generalized solutions. Positive definite case
• 1.1. Preliminaries
• 1.2. Uniqueness and existence theorems
• 2. Degenerate case
• 2.1. Preliminaries
• 2.2. Solvability theorems. The case of a bounded interval
• 2.3. Solvability theorems. The case of the interval (0, 8)
• 2.4. Smoothness of solutions. Orthogonality conditions
• 2.5. The periodic problem. Linear inverse problems
• 3. The Fourier method
• 3.1. Representation of solutions. First order equations
• 3.2. Some problems for the second order equations
• 4. Some applications to partial differential equations
• 4.1. Higher order parabolic equations with changing time direction
• 4.2. Second order parabolic equations with changing time direction
• 4.3. Orthogonality conditions. Parabolic equations
• 4.4. Second order mixed type equations. Orthogonality conditions
• Bibliography
• Index