Matrix Analysis and Entrywise Positivity Preservers

Part I. Preliminaries, Entrywise Powers Preserving Positivity in Fixed Dimension: 1. The cone of positive semidefinite matrices; 2. The Schur product theorem and nonzero lower bounds; 3. Totally positive (TP) and totally non-negative (TN) matrices; 4. TP matrices - generalized Vandermonde and Hankel moment matrices; 5. Entrywise powers preserving positivity in fixed dimension; 6. Mid-convex implies continuous, and 2 x 2 preservers; 7. Entrywise preservers of positivity on matrices with zero patterns; 8. Entrywise powers preserving positivity, monotonicity, superadditivity; 9. Loewner convexity and single matrix encoders of preservers; 10. Exercises; Part II. Entrywise Functions Preserving Positivity in All Dimensions: 11. History - Shoenberg, Rudin, Vasudeva, and metric geometry; 12. Loewner's determinant calculation in Horn's thesis; 13. The stronger Horn-Loewner theorem, via mollifiers; 14. Stronger Vasudeva and Schoenberg theorems, via Bernstein's theorem; 15. Proof of stronger Schoenberg Theorem (Part I) - positivity certificates; 16. Proof of stronger Schoenberg Theorem (Part II) - real analyticity; 17. Proof of stronger Schoenberg Theorem (Part III) - complex analysis; 18. Preservers of Loewner positivity on kernels; 19. Preservers of Loewner monotonicity and convexity on kernels; 20. Functions acting outside forbidden diagonal blocks; 21. The Boas-Widder theorem on functions with positive differences; 22. Menger's results and Euclidean distance geometry; 23. Exercises; Part III. Entrywise Polynomials Preserving Positivity in Fixed Dimension: 24. Entrywise polynomial preservers and Horn-Loewner type conditions; 25. Polynomial preservers for rank-one matrices, via Schur polynomials; 26. First-order approximation and leading term of Schur polynomials; 27. Exact quantitative bound - monotonicity of Schur ratios; 28. Polynomial preservers on matrices with real or complex entries; 29. Cauchy and Littlewood's definitions of Schur polynomials; 30. Exercises.