Beschreibung
The investigation of many mathematical problems is significantly simplified if it is possible to reduce them to equations involving continuous or com pletely continuous operators in function spaces. In particular, this is true for non-linear boundary value problems and for integro-differential and integral equations. To effect a transformation to equations with continuous or completely continuous operators, it is usually necessary to reduce the original problem to one involving integral equations. Here, negative and fractional powers of those unbounded differential operators which constitute 'principal parts' of the original problem, are used in an essential way. Next there is chosen or constructed a function space in which the corresponding integral oper ator possesses sufficiently good properties. Once such a space is found, the original problem can often be analyzed by applying general theorems (Fredholm theorems in the study of linear equations, fixed point principles in the study of non-linear equations, methods of the theory of cones in the study of positive solutions, etc.). In other words, the investigation of many problems is effectively divided into three independent parts: transformation to an integral equation, investi gation of the corresponding integral expression as an operator acting in function spaces, and, finally, application of general methods of functional analysis to the investigation of the linear and non-linear equations.
Autorenporträt
Inhaltsangabe1. Linear operators in L? spaces.- 1. The space L?.- 1.1. Description of the spaces.- 1.2. Criteria for compactness.- 1.3. Continuous linear functionals and weak convergence.- 1.4. Semi-ordering in the spaces S and L?.- 1.5. Projections and bases of Haar type.- 1.6. Operators in the spaces L?.- 2. Continuous linear operators.- 2.1. Linear operators.- 2.2. Regular operators.- 2.3. The M. Riesz interpolation theorem.- 2.4. Interpolation theorems for regular operators.- 2.5. Classes of L-characteristics of linear operators.- 2.6. On a property of regular operators.- 2.7. The Marcinkiewics interpolation theorem.- 3. Compact linear operators.- 3.1. Compact linear operators.- 3.2. Compactness and adjoint operators.- 3.3. Properties of operators compact in measure.- 3.4. Interpolation properties of compactness.- 3.5. Strongly continuous linear operators.- 2. Continuity and compactness of linear integral operators.- 4. General theorems on continuity on integral operators.- 4.1. Linear integral operators.- 4.2. Regular operators.- 4.3. Example of a non-regular operator.- 4.4. The adjoint operator.- 4.5. Operators with symmetric kernels.- 4.6. Products of integral operators.- 4.7. Truncations of kernels of integral operators.- 5. General theorems on compactness of integral operators.- 5.1. Problem setting.- 5.2. Regular operators acting from Lo to L?0 and from L?0 to L1.- 5.3. Regular operators acting from L?0 to L?0 where 0 0).- 6.7. Integral operators acting from L? To c.- 6.8. ?0-Cobounded linear operators.- 6.9. Compactness of ?0-cobounded operators.- 6.10. Interpolation properties of u0-boundedness.- 6.11. On weakly compact operators in l1.- 7. Integral operators with kernels satisfying conditions of kantorovic type.- 7.1. Simplest criteria.- 7.2. Theorems with intermediate conditions.- 7.3. Lemmas.- 7.4. Applications of theorems on adjoint operators.- 7.5. Fundamental theorems.- 7.6. Conditions of 'Kantorovic' type.- 7.7. Summability of kernels of integral operators.- 8. Operators of potential type.- 8.1. Definitions.- 8.2. Simplest theorems on continuity and compactness of potentials.- 8.3. Interpolation theorem of Stein-Weiss.- 8.4. Limit theorems on continuity of potentials.- 8.5. Operators of potential type.- 8.6. The logarithmic potential.- 8.7. Iterates of operators of potential type.- 8.8. Generalizations to the case of distinct dimensions.- 8.9. Potentials with respect to non-Lebesgue measure.- 3. Fractional powers of selfadjoint operators.- 9. Splitting of linear operators.- 9.1. Square root of selfadjoint operators.- 9.2. Splitting of an operator.- 9.3. L-Characteristic of a square root.- 9.4. Representation of compact operators.- 9.5. Square root of integral operator.- 9.6. Example.- 9.7. Investigation of integral operators by means of properties of iterated kernels.- 9.8. Remark on Mercers' theorem.- 10. Fractional powers of bounded operators.- 10.1. The spectral function.- 10.2. Fractional powers of bounded selfadjoint operators.- 10.3. The fundamental theorem.- 10.4. Operators in real spaces.- 10.5. Fractional powers of compact operators.- 10.6. L-Characteristics of fractional powers of operators.- 10.7. Fractional powers of integral operators.- 11. Unbounded selfadjoint operators.- 11.1. Closed operators.- 11.2.
