Variational principles for spectral radii of positive functional operators

A. B. Antonevich

Contemporary Mathematics. Fundamental Directions, Proceedings of the Fourth International Conference on Differential and Functional-Differential Equations (Moscow, August 14–21, 2005). Part 1, Tome 15 (2006), pp. 5-18.

Résumé

Functional operators, i.e., sums of weighted shift operators generated by various maps, are considered. For functional operators with positive coefficients, variational principles for spectral radii are obtained. These principles say that the logarithm of the spectral radius is the Legendre transform of a certain convex functional $T$ defined on the set of probability vector-valued measures and depending on the original dynamical system and the functional space considered. In the subexponential case, we obtain the combinatorial structure of the functional $T$ with the help of the corresponding random walk process constructed according to the dynamical system.

Bibliographie

[1] Antonevich A. B., “Usloviya obratimosti operatorov s vypukloi ratsionalno nezavisimoi sistemoi sdvigov”, DAN SSSR, 256:1 (1981), 11–14 | MR | Zbl

[2] Antonevich A. B., Lineinye funktsionalnye uravneniya: operatornyi podkhod, Universitetskoe, Minsk, 1988 | MR | Zbl

[3] Antonevich A. B., Bakhtin V. I., Lebedev A. V., “Variatsionnyi printsip dlya spektralnogo radiusa operatorov vzveshennogo sdviga v lebegovskikh prostranstvakh”, Tr. Instituta matematiki NAN Belarusi, 5, 2000, 13–17 | Zbl

[4] Antonevich A. B., Zaikovskii K., “Variatsionnyi printsip dlya spektralnogo radiusa modelnogo funktsionalnogo operatora”, Tr. Instituta matematiki NAN Belarusi, 12, no. 2, 2004, 18–25

[5] Gantmakher F. R., Teoriya matrits, Nauka, M., 1967 | MR

[6] Grigorchuk R. I., “Simmetricheskie sluchainye bluzhdaniya na diskretnykh gruppakh”, Mnogokomponentnye sluchainye sistemy, Nauka, M., 1978, 132–152 | MR

[7] Dorogov V. I., Chistyakov V. P., Veroyatnostnye modeli prevrascheniya chastits, Nauka, M., 1988 | MR

[8] Katok A. B., Khasselblat B., Vvedenie v sovremennuyu teoriyu dinamicheskikh sistem, Faktorial, M., 1999

[9] Kornfeld I. P., Sinai Ya. G., Fomin S. V., Ergodicheskaya teoriya, Nauka, M., 1980 | MR

[10] Latushkin Yu. D., Stepin A. M., “Operatory vzveshennogo sdviga na topologicheskoi markovskoi tsepi”, Funkts. analiz i ego prilozh., 22:4 (1988), 86–87 | MR

[11] Maslov V. P., “Statisticheskii ansambl i kvantovanie termodinamiki”, Mat. zametki, 71:4 (2002), 558–566 | MR | Zbl

[12] Feller V., Vvedenie v teoriyu veroyatnostei i ee prilozheniya, t. 1, Mir, M., 1984

[13] Abramovich Y. A., Arenson E. L., Kitover A. K., Banach $C(K)$-modules and operators preserving disjointness, Longman Scientific and Technical, Harlow–Essex, 1992 | MR | Zbl

[14] Antonevich A., Bakhtin V., Lebedev A., “Thermodynamics and spectral radius”, Nonlinear Phenomena in Complex Systems, 4:4 (2001), 318–321 | MR

[15] Antonevich A., Lebedev A., Functional differential equations. I: $C^*$-theory, Longman, Harlow, 1994 | Zbl

[16] Chen D. R., “Spectral radii and eigenvalues of subdivision operator”, Proc. Am. Math. Soc., 132 (2004), 1113–1123 | DOI | MR | Zbl

[17] Chicone C., Latushkin Yu., Evolution semigroup in dynamical systems and differential equations, AMS, Providence, RI, 1999 | MR | Zbl

[18] Danes J., “On the local spectral radius”, Cas. Pest. Mat., 112 (1987), 177–187 | MR | Zbl

[19] Ekeland I., Temam R., Convex analysis and variational problems, North-Holland Publ., Amsterdam, 1976 | MR | Zbl

[20] Kesten H., “Symmetric random walks on groups”, Trans. Am. Math. Soc., 92 (1959), 336–354 | DOI | MR | Zbl

[21] Kravchenko V. G., Litvinchuk G. S., Introduction to the theory of singular integral operators with shift, Kluwer Acad. Publ., Dordrecht, 1994 | MR

[22] Skubachevskii A. L., Elliptic Functional differential equations and applications, Birkhäuser, Basel, Boston, Berlin, 1997 | MR | Zbl