Bounded Integral Operators on L 2 Spaces

The subject. The phrase "integral operator" (like some other mathematically informal phrases, such as "effective procedure" and "geometric construction") is sometimes defined and sometimes not. When it is defined, the definition is likely to vary from author to author. While the definition almost always involves an integral, most of its other features can vary quite considerably. Superimposed limiting operations may enter (such as L2 limits in the theory of Fourier transforms and principal values in the theory of singular integrals), IJ' spaces and abstract Banach spaces may intervene, a scalar may be added (as in the theory of the so-called integral operators of the second kind), or, more generally, a multiplication operator may be added (as in the theory of the so-called integral operators of the third kind). The definition used in this book is the most special of all. According to it an integral operator is the natural "continuous" generali­ zation of the operators induced by matrices, and the only integrals that appear are the familiar Lebesgue-Stieltjes integrals on classical non-pathological mea­ sure spaces. The category. Some of the flavor of the theory can be perceived in finite­ dimensional linear algebra. Matrices are sometimes considered to be an un­ natural and notationally inelegant way of looking at linear transformations. From the point of view of this book that judgement misses something.

§1. Measure Spaces.- Example 1.1. Separable, not ?-finite.- Example 1.2. Finite, not separable.- §2. Kernels.- § 3. Domains.- Example 3.1. Domain 0.- Example 3.2. Hilbert transform.- Problem 3.3. Closed domain.- Example 3.4. Dense domain.- Example 3.5. Dense domain.- Example 3.6. Non-closed kernel.- Example 3.7. Non-closed kernel.- Theorem 3.8. Carleman kernels.- Lemma 3.9. Dominated subsequences.- Theorem 3.10. Full domain.- Example 3.11. Everywhere defined kernels.- Problem 3.12. Closed domains and kernels.- §4. Boundedness.- Lemma 4.1. Square integrable kernels.- Example 4.2. Dyads.- Lemma 4.3. Rank 1.- Corollary 4.4. Finite rank.- Theorem 4.5. Hilbert-Schmidt operators.- Corollary 4.6. Compactness.- Corollary 4.7. Singular values.- §5. Examples.- Example 5.1. Inflated identity.- Theorem 5.2. Schur test.- Example 5.3. Abel kernel.- Example 5.4. Cesàro kernel.- Example 5.5. Hilbert-Hankel matrix.- Theorem 5.6. Toeplitz matrices.- Example 5.7. Hilbert-Toeplitz matrix.- Example 5.8. Discrete Fourier transform.- §6. Isomorphisms.- Theorem 6.1. Induced unitary operators.- Theorem 6.2. Transforms of kernels.- Corollary 6.3. Unitary equivalence.- Corollary 6.4. Preservation of structure.- Example 6.5. Projection on L2(II).- Example 6.6. Atomic spaces versus ?.- §7. Algebra.- Problem 7.1. Multipliability.- Example 7.2. Compact Fourier transform.- Theorem 7.3. Operators on atomic spaces.- Lemma 7.4. Integrable approximation.- Theorem 7.5. Conjugate transposes.- Corollary 7.6. Atomic domain.- Corollary 7.7. Matrices.- §8. Uniqueness.- Theorem 8.1. Uniqueness.- Problem 8.2. Determination.- Example 8.3. Non-measurable kernel.- Problem 8.4. Measurability.- Theorem 8.5. Identity operator.- Theorem 8.6. Multiplication operators.- §9. Tensors.- Theorem 9.1. Direct sums.- Corollary 9.2. Carleman kernels.- Theorem 9.3. Tensor products.- Problem 9.4. Bounded kernels.- Theorem 9.5. Tensor multiplicativity of Int.- Theorem 9.6. Tensors with dyads.- Example 9.7. Isometry on L2(II).- Example 9.8. Inflations as tensor products.- Theorem 9.9. Bounded matrices.- Corollary 9.10. Schur products.- Example 9.11. Schur products with dyads.- §10. Absolute Boundedness.- Example 10.1. Hilbert-Toeplitz matrix.- Example 10.2. Discrete Fourier transform.- Example 10.3. Direct sum matrix.- Example 10.4. Divisible spaces.- Theorem 10.5. Characterization.- Corollary 10.6. Adjoints.- Theorem 10.7. Products.- Theorem 10.8. Non-invertibility.- Theorem 10.9. Schur products.- Example 10.10. Unbounded Schur products.- Remark 10.11. Tensor quotients.- §11. Carleman Kernels.- Example 11.1. Absolutely bounded, not Carleman.- Theorem 11.2. Inclusion relations.- Example 11.3. Counterexamples.- Theorem 11.4. Strong boundedness.- Theorem 11.5. Carleman functions.- Theorem 11.6. Right ideal.- Corollary 11.7. Non-invertibility.- Problem 11.8. Right ideal.- Theorem 11.9. Co-boundedness.- Theorem 11.10. Hermitian kernels.- Theorem 11.11. Normal Carleman adjoints.- Problem 11.12. Normal integral adjoints.- Example 11.13. Non-Carleman integral adjoint.- §12. Compactness.- Lemma 12.1. Convolution kernels on L1.- Theorem 12.2. Convolution kernels on L2.- Corollary 12.3. Compactness.- Example 12.4. Non-integral, compact.- §13. Compactness.- Lemma 13.1. Large characteristic functions.- Lemma 13.2. Absolute continuity.- Example 13.3. Non-absolute continuity.- Lemma 13.4. Hille-Tamarkin kernels.- Example 13.5. Non-Hille-Tamarkin kernels.- Remark 13.6. Hille-Tamarkin operators.- Lemma 13.7. Integrable kernels.- Theorem 13.8. compactness.- Corollary 13.9. Hilbert-Schmidt approximation.- § 14. Essential Spectrum.- Example 14.1. Tensor products and spectra.- Theorem 14.2. Atkinson’s theorem.- Theorem 14.3. Normal operators.- Theorem 14.4. A and A*A.- Corollary 14.5. A and AA*.- Theorem 14.6. Orthonormal sequences, left.- Corollary 14.7. Orthonormal sequences, right.- Remark 14.8. Absolute boundedness and invertibility.- Remark 14.9. Non-emptiness.- Theorem 14.10. Normal Carleman operators.- Lemma 14.11. Nearly invariant subspaces.- Remark 14.12. Hilbert-Schmidt strengthening.- Theorem 14.13. Weyl-von Neumann theorem.- Problem 14.14. Normal generalization.- Problem 14.15. Quasidiagonal generalization.- §15. Characterization.- Theorem 15.1. Integral operator, essential spectrum.- Remark 15.2. Right versus left.- Corollary 15.3. Unitary transforms.- Lemma 15.4. Matrix inflations.- Remark 15.5. Partially atomic spaces.- Lemma 15.6. Perturbations of Hermitian operators.- Theorem 15.7. Carleman operator, essential spectrum.- Corollary 15.8. Carleman if and only if integral.- Example 15.9. Unilateral shift.- Example 15.10. Non-simultaneity of A and A*.- Theorem 15.11. Simultaneity of A and A*.- Corollary 15.12. Simultaneous integral representability.- Lemma 15.13. Large 0 direct summand.- Theorem 15.14. Simultaneous Carleman representability.- Corollary 15.15. Simultaneous Carleman if and only if integral.- Problem 15.16. Absolutely bounded operators.- Theorem 15.17. Essential non-invertibility of A*A+AA*.- Theorem 15.18. Absolutely bounded operators.- §16. Universality.- Theorem 16.1. Universal integral operators.- Remark 16.2. Universal Carleman operators.- Problem 16.3. Small unitary transforms.- Lemma 16.4. Operator norm.- Theorem 16.5. Universally absolutely bounded matrices.- §17. Recognition.- Remark 17.1. Pointwise domination.- Theorem 17.2. Carleman characterization.- Corollary 17.3. Hilbert-Schmidt characterization.- Problem 17.4. Integral characterization.- Theorem 17.5. Orthonormal Carleman characterization.- Problem 17.6. Orthonormal integral characterization.- Theorem 17.7. Null-sequence Carleman characterization.- Appendix A. Finiteness and Countability Conditions.- Appendix B. Pointwise Unbounded Bounded Kernels.- Theorem B1. Pointwise unbounded subkernels.- Corollary B2. Subrectangles.- Corollary B3. Square integrable kernels.- Problem B4. Unbounded subkernels.- Appendix C. Riemann-Lebesgue Lemma.- Notes.- References.