Francesco Altomare - the remarkable mathematician and human being

Abstract

We laconically describe the great contributions of Professor Francesco Altomare to mathematical research and PhD education, and his unique status in the mathematical community. In particular, we present and give examples of his innovative and great achievements related to the following areas of mathematics: Functional Analysis, Operator Theory, Potential Theory, Approximation Theory, Probability Theory, Function Spaces, Choquet's Theory, Dirichlet's Problem and Semigroup Theory. Moreover, we report on and give concrete examples of his unique way to work together with PhD students, both before and sometimes also after their dissertation. Finally, we shortly describe his remarkable “class travel” from “simple” conditions with no academic traditions in his family in the small hometown Giovinazzo to finally become the broad, ingenious, and powerful mathematician he is regarded to be today.

Keywords

• Francesco Altomare
• History of Mathematics
• Functional Analysis
• Operator Theory
• Potential Theory
• Approximation Theory
• Probability Theory
• Function Spaces
• Choquet’s Theory
• Dirichlet’s Problem
• Semigroup Theory
• Evolution Equations
• Ph.D education

References

1. F. Altomare: Proiettori positivi, famiglie risolventi e problema di Dirichlet, Ricerche Mat., 26 (1) (1977), 63–78.
2. F. Altomare: Operatori di Lion generalizzati e famiglie risolventi, Boll. Un. Mat. Ital. B (5), 15 (1) (1978), 60–79.
3. F. Altomare: Lion operators over the product of compact spaces, semigroups of positive operators, and the Dirichlet problem, Ricerche Mat., 27 (1) (1978), 33–58.
4. F. Altomare: Théorèmes de convergence de type Korovkin relativement a une applications lineaire positive, Boll. Un. Mat. Ital. B (5), 16 (3) (1979), 1013–1031.
5. F. Altomare: On the universal convergence sets, Ann. Mat. Pura Appl., 138 (4) (1984), 223–243.
6. F. Altomare: Positive linear forms and their determining subspaces, Ann. Math. Pura Appl., 154 (4) (1989), 243–258.
7. F. Altomare: Limit semigroups of Bernstein-Schnabl operators associated with positive projections, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 16 (2) (1989), 259–279.
8. F. Altomare: Asymptotic formulae for Bernstein-Schnabl operators and smoothness, Bollettino U.M.I., II (9) (2009), 135–150, Corrigendum: Bollettino U.M.I., IV (9) (2011), 259–262.

9. F. Altomare: Korovkin-type theorems and approximation by positive linear operators, Surv. Approx. Theory, 5 (2010), 92–164.
10. F. Altomare: Iterates of Markov operators and constructive approximation of semigroups, Constr. Math. Anal., 2 (1) (2019), 22–39.
11. F. Altomare: On the convergence of sequences of positive linear operators and functionals on bounded function spaces, Proc. Amer. Math. Soc., to appear. DOI: https://doi.org/10.1090/proc/15445.
12. F. Altomare et al: The Maratea Conferences on Functional Analysis and Approximation Theory from 1989 to 2009-An overview, 2009, free available on line at https://galileo.dm.uniba.it/Members/altomare/Maratea-Conferences.pdf.
13. F. Altomare, R. Amiar: Asymptotic formulae for positive linear operators, Mathematica Balkanica New Series, 16 (2002), Fasc. 1–4, 283–304.
14. F. Altomare, M. Campiti: Korovkin-type approximation theory and its applications, De Gruyter Studies in Mathematics, 17, Walter de Gruyter & C., Berlin (1994).
15. F. Altomare, M. Cappelletti Montano: Affine projections on adapted subalgebras of continuous functions, Positivity, 9 (4) (2005), 625–643.
16. F. Altomare, M. Cappelletti Montano: Regular vector lattices of continuous functions and Korovkin-type theorems - Part I, Studia Math., 171 (3) (2005), 239–260.
17. F. Altomare, M. Cappelletti Montano: On some density theorems in regular vector lattices of continuous functions, Collect. Math., 58 (2) (2007), 131–149.
18. F. Altomare, M. Cappelletti Montano, V. Leonessa and I. Ra¸sa: On differential operators associated with Markov operators, J. Funct. Anal., 266 (6) (2014), 3612–3631.
19. F. Altomare, M. Cappelletti Montano, V. Leonessa and I. Rasa: On Markov operators preserving polynomials. J. Math. Anal. Appl., 415 (1) (2014), 477–495.
20. F. Altomare, M. Cappelletti Montano, V. Leonessa and I. Ra¸sa: Markov operators, positive semigroups and approximation processes, De Gruyter Studies in Mathematics 61, De Gruyter, Berlin (2014).
21. F. Altomare, M. Cappelletti Montano, V. Leonessa and I. Ra¸sa: Elliptic differential operators and positive semigroups associated with generalized Kantorovich operators, J. Math. Anal. Appl., 458 (1) (2018), 153–173.
22. F. Altomare, S. Diomede: Asymptotic formulae for positive linear operators: direct and converse results, Jaén J. Approx., 2 (2) (2010), 255–287.
23. F. Altomare, V. Leonessa and I. Ra¸sa: On Bernstein-Schnabl operators on the unit interval, Z. Anal. Anw., 27 (3) (2008), 353–379.
24. F. Altomare, V. Leonessa: An invitation to the study of evolution equations by means of positive linear operators, Lecture Notes of Seminario Interdisciplinare di Matematica, 8 (2009), 1–41.
25. F. Altomare, S. Milella and G. Musceo: On a class of positive C0-semigroups on weighted continuous function spaces, Note Mat., 31 (1) (2011), 15–27.
26. F. Altomare, S. Milella and G. Musceo: Multiplicative perturbations of the Laplacian and related approximation problems, J. Evol. Equ., 11 (2011), 771–792.
27. F. Altomare, I. Ra¸sa: Towards a characterization of a class of differential operators associated with positive projection, Atti Sem. Mat. Fis. Univ. Modena, 46 (1998), suppl., 3–38.
28. M. Campiti: On the scientific heritage of Professor Francesco Altomare. A presentation of some meaningful results, Recent developments in Functional Analysis and Approximation Theory, Lecce (Italy) (2011), unpublished.
29. A. Pietsch: History of Banach spaces and linear operators, Birkhäuser Boston, Inc., Boston, MA (2007).
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