Operator Theory

Introduction. CHAPTER 1. HOLOMORPHIC FREDHOLM OPERATOR FUNCTIONS. I.I. Lifting and open mapping theorems. 1.2. Some classes of linear operators. 1.3. Banach vector bundles. 1.4. Fredholm operators that depend continuously on a paxameter. 1.5. Some information from complex analysis. A. Interpolation of entire functions of finite order. B. Some information from the complex analysis in several vaxiables. C. Some problems of infinite-dimensional complex analysis. 1.6. Fredholm operators that depend holomorphically on a parameter. 1.7. Image and cokernel of a Fredholm morphism in spaces of holomorphic sections. 1.8. Image and cokernel of a Fredholm morphism in spaces of holomorphic sections with bounds. 1.9. Comments and references. CHAPTER 2. SPACES, OPERATORS AND TRANSFORMS. 2.1. Basic spaces and operators. 2.2. Fourier transform on the group of periods. 2.3. Comments and references. CHAPTER 3. FLOQUET THEORY FOR HYPOELLIPTIC EQUATIONS AND SYSTEMS IN THE WHOLE SPACE. 3.1. Floquet - Bloch solutions. Quasimomentums and Floquet exponents. 3.2. Floquet expansion of solutions of exponential growth. 3.3. Completeness of Floquet solutions in a class of solutions of faster growth. 3.4. Other classes of equations. A. Elliptic systems. B. Hypoelliptic equations and systems. C. Pseudodifferential equations. D. Smoothness of coefficients. 3.5. Comments and references. CHAPTER 4. PROPERTIES OF SOLUTIONS OF PERIODIC EQUATIONS. 4.1. Distribution of quasimomentums and decreasing solutions. 4.2. Solvability of non-homogeneous equations. 4.3. Bloch property. 4.4. Quasimomentum dispersion relation. Bloch vaxiety. 4.5. Some problems of spectral theory. 4.6. Positive solutions. 4.7 Comments and references. CHAPTER 5. EVOLUTION EQUATIONS. 5.1. Abstract hypoelliptic evolution equations on the whole axis. 5.2. Some degenerate cases. 5.3. Cauchy problem for abstract parabolic equations. 5.4. Elliptic and parabolic boundary value problems in a cylinder. A. Elliptic problems. B. Paxabolic problems. 5.5. Comments and references. CHAPTER 6. OTHER CLASSES OF PROBLEMS. 6.1. Equations with deviating arguments. 6.2. Equations with coefficients that do not depend on some arguments. 6.3. Invariant differential equations on Riemannian symmetric spaces of non-compact type. 6.4. Comments and references. Bibliography. Index of symbols. Index.