Kernel theorems and nuclearity in idempotent mathematics. An algebraic approach

G. L. Litvinov   ; G. B. Shpiz

Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial and algoritmic methods. Part XIV, Tome 331 (2006), pp. 60-83

Résumé

In the framework of idempotent mathematics, some analogs for the well-known kernel theorems of L. Schwartz and A. Grothendieck are examined. Idempotent versions of nuclear spaces (in the sense of A. Grothendieck) are described. An algebraic approach is used, so topological concepts and results are simulated by means of algebraic tools.

Bibliographie

[1] V. P. Maslov, Asimptoticheskie metody resheniya psevdodifferentsialnykh uravnenii, Nauka, M., 1987 | MR

[2] V. P. Maslov, Méthodes opératorielles, Mir, Moscow, 1987 | MR | Zbl

[3] V. P. Maslov, V. N. Kolokoltsov, Idempotentnyi analiz i ego primenenie v optimalnom upravlenii, Nauka, M., 1994 | MR

[4] V. N. Kolokoltsov, V. P. Maslov, Idempotent Analysis and Applications, Kluwer Acad. Publ., Dordrecht, 1997 | MR | Zbl

[5] V. P. Maslov, S. N. Samborski\u i (eds.), Idempotent Analysis, Adv. Sov. Math., 13, Amer. Math. Soc., Providence, RI, 1992 | MR

[6] J. Gunawardena (ed.), Idempotency, Cambridge Univ. Press, Cambridge, 1998 | MR

[7] G. L. Litvinov, V. P. Maslov, Correspondence Principle for Idempotent Calculus and Some Computer Applications, IHES/M/95/33 Institut des Hautes Études Scientifiques, Bures-sur-Yvette, 1995, see also [6, pp. 420–443] ; arXiv: /math.GM/0101021 | MR

[8] G. L. Litvinov, V. P. Maslov, G. B. Shpiz, “Idempotentnyi funktsionalnyi analiz. Algebraicheskii podkhod”, Mat. zametki, 69:5 (2001), 696–729 ; arXiv: /math.FA/0009128 | MR | Zbl

[9] G. L. Litvinov, V. P. Maslov (eds.), Idempotent Mathematics and Mathematical Physics, Contemp. Math., 377, Providence, RI, 2005 | MR

[10] G. L. Litvinov, “Dekvantovanie Maslova, idempotentnaya i tropicheskaya matematika: kratkoe Vvedenie”, Zap. nauchn. semin. POMI, 326, 2005, 145–182 ; arXiv: /math.GM/0507014 | MR | Zbl

[11] G. L. Litvinov, V. P. Maslov, G. B. Shpiz, “Tenzornye proizvedeniya idempotentnykh polumodulei. Algebraicheskii podkhod”, Mat. zametki, 65:4 (1999), 573–586 ; arXiv: /math.FA/0101153 | MR | Zbl

[12] G. L. Litvinov, V. P. Maslov, G. B. Shpiz, “Lineinye funktsionaly na idempotentnykh prostranstvakh. Algebraicheskii podkhod”, Dokl. Akad. Nauk, 363:3 (1998), 298–300 ; arXiv: /math.FA/0012268 | MR | Zbl

[13] V. N. Kolokoltsov, V. P. Maslov, “Obschii vid endomorfizmov v prostranstve nepreryvnykh funktsii so znacheniem v chislovom kommutativnom polukoltse (s operatsiei $\oplus=\max$)”, Dokl. Akad. Nauk SSSR, 295:2 (1987), 283–287 | MR

[14] V. N. Kolokoltsov, “Idempotent structures in optimization”, J. Math. Sci., 104:1 (2001), 847–880 | DOI

[15] M. Akian, S. Gaubert, V. Kolokoltsov, “Set coverings and invertibility of functional Galois connections”, Idempotent Mathematics and Mathematical Physics, Contemporary Mathematics, 377, American Mathematical Society, Providence, RI, 2005, 19–52 | MR

[16] G. Cohen, S. Gaubert, J.-P. Quadrat, “Duality and separation theorems in idempotent semimodules”, Linear Algebra Appl., 379 (2004), 395–422 ; arXiv: /math.FA/0212294 | DOI | MR | Zbl

[17] O. Viro, “Dequantization of real algebraic geometry on a logarithmic paper”, 3rd European Congress of Mathematics, Barcelona, 2000; arXiv: /math.AG/0005163

[18] G. L. Litvinov, A. N. Sobolevski\u i, “Exact interval solutions of the discrete Bellman equation and polynomial complexity of problems in interval idempotent linear algebra”, Dokl. Math., 62:2 (2000), 199–201 ; arXiv: /math.LA/0101041 | MR | Zbl

[19] G. L. Litvinov, A. N. Sobolevski\u i, “Idempotent interval analysis and optimization problems”, Reliable Computing, 7:5 (2001), 353–377 ; arXiv: /math.SC/0101080 | DOI | MR | Zbl

[20] L. Schwartz, “Théorie des noyaux”, Proceedings of the International Congress of Mathematicians, 1 (Cambridge, Mass., Aug. 30-Sept. 6, 1950), 1, 1952, 220–230 | MR | Zbl

[21] A. Grothendieck, Produits tensorielle topologiques et espace nucléairs, Mem. Amer. Math. Soc., 16, Providence, RI, 1955 | MR

[22] Kh. Shefer, Topologicheskie vektornye prostranstva, Mir, M., 1971 | MR

[23] P. I. Dudnikov, S. N. Samborskii, “Endomorfizmy polumodulei nad polukoltsami s idempotentnoi operatsiei”, Izvestiya AN SSSR, ser. matem., 55:1 (1991), 93–109 | MR

[24] M. A. Shubin, Algebraicheskie zamechaniya ob idempotentnykh polukoltsakh i teorema o yadre v prostranstvakh ogranichennykh funktsii, Institut novykh tekhnologii, M., 1990 | Zbl

[25] I. Singer, “Some relations between linear mappings and conjugations in idempotent analysis”, J. Math. Sci., 115:5 (2003), 2610–2630 | DOI | MR | Zbl

[26] G. Choquet, “Theory of capacities”, Ann. Inst. Fourier, 5 (1955), 131–295 | MR

[27] W. Bryc, Diffusion Processes and Related Problems in Analysis, ed. M. Pynsky, Birkhäuser, 1990, 447–472 | MR

[28] A. Puhalskii, “Large deviations of semimartingales via convergence of the predictable characteristics”, Stochastics, 49 (1994), 27–85 | MR | Zbl

[29] V. Breier, O. Gulinskii, Bolshie ukloneniya na beskonechnomernykh prostranstvakh, Preprint, MFTI, M., 1996

[30] O. V. Gulinsky, The principle of the largest terms and large deviations for a class of nonlinear functionals, Preprint, 2001

[31] J. Gunawardena, “An introduction to idempotency”, Idempotency, Cambridge University Press, Cambridge, 1998, 1–49 | MR | Zbl

[32] G. L. Litvinov, G. B. Shpiz, “Yadernye polumoduli i teoremy o yadre v idempotentnom analize. Algebraicheskii podkhod”, Dokl. Akad. Nauk, 386:3 (2002), 300–303 ; arXiv: /math.FA/0206026 | MR | Zbl

[33] G. Birkgof, Teoriya reshetok, Nauka, M., 1984 | MR