Maps with separable dynamics and the spectral properties of the operators generated by them

A. B. Antonevich   ; A. A. Ahmatova   ; Ju. Makowska

Sbornik. Mathematics, Tome 206 (2015) no. 3, pp. 341-369

Résumé

A map $\alpha $ of a space $X$ into itself generates weighted shift operators $B$ in function spaces on $X$. The spectral properties of such operators are intimately connected with the dynamics of $\alpha$. It was known previously that the spectrum of an operator depends only on the set of invariant ergodic measures for $\alpha$. Conditions for the right invertibility of the operators $B-\lambda I$ are obtained when $\lambda$ is a spectral value. The main result states that right invertibility is only possible when a nontrivial attractor exists. Bibliography: 29 titles.

Détail

Keywords: spectrum of an operator, one-sided invertibility, essential spectrum, attractor, ergodic measure.

Bibliographie

[1] P. R. Halmos, Lectures on ergodic theory, Publications of the Mathematical Society of Japan, 3, Math. Soc. Japan, Tokyo, 1956, vii+99 pp. | MR | Zbl | Zbl

[2] I. U. Bronshtein, Neavtonomnye dinamicheskie sistemy, Shtiintsa, Kishinev, 1984, 292 pp. | MR

[3] A. Katok, B. Hasselblatt, Introduction to the modern theory of dynamical systems, Encyclopedia Math. Appl., 54, Cambridge Univ. Press, Cambridge, 1995, xviii+802 pp. | DOI | MR | Zbl

[4] J. Hale, Theory of functional differential equations, Appl. Math. Sci., 3, 2nd ed., Springer-Verlag, New York–Heidelberg, 1977, x+365 pp. | MR | MR | Zbl | Zbl

[5] Linear functional equations. Operator approach, Oper. Theory Adv. Appl., 83, Birkhäuser Verlag, Basel, 1996, viii+179 pp. | DOI | MR | MR | Zbl | Zbl

[6] A. Antonevich, A. Lebedev, Functional-differential equations. I. $C^*$-theory, Pitman Monogr. Surveys Pure Appl. Math., 70, Longman Sci. Tech., Harlow, 1994, viii+504 pp. | MR | Zbl

[7] A. L. Skubachevskii, Elliptic functional differential equations and applications, Oper. Theory Adv. Appl., 91, Birkhäuser Verlag, Basel, 1997, x+293 pp. | MR | Zbl

[8] C. Chicone, Yu. Latushkin, Evolution semigroups in dynamical systems and differential equations, Math. Surveys Monogr., 70, Amer. Math. Soc., Providence, RI, 1999, x+361 pp. | DOI | MR | Zbl

[9] V. G. Kravchenko, G. S. Litvinchuk, Introduction to the theory of singular integral operators with shift, Transl. from the Russian manuscript by Litvinchuk, Math. Appl., 289, Kluwer Acad. Publ., Dordrecht, 1994, xvi+288 pp. | DOI | MR | Zbl

[10] S. Stević, R. Chen, Z. Zhou, “Weighted composition operators between Bloch-type spaces in the polydisc”, Sb. Math., 201:2 (2010), 289–319 | DOI | DOI | MR | Zbl

[11] A. N. Sharkovsky, Yu. L. Maistrenko, E. Yu. Romanenko, Difference equations and their applications, Math. Appl., 250, Kluwer Acad. Publ., Dordrecht, 1993, xii+358 pp. | DOI | MR | MR | Zbl | Zbl

[12] R. Mardiev, “Kriterii poluneterovosti odnogo klassa singulyarnykh integralnykh operatorov s nekarlemanovskim sdvigom”, Dokl. AN UzSSR, 2:2 (1985), 5–7 | MR | Zbl

[13] G. Belitskii, Yu. I. Lyubich, “On the normal solvability of cohomological equations on compact topological spaces”, Recent progress in operator theory (Regensburg, 1995), Oper. Theory Adv. Appl., 103, Birkhäuser, Basel, 1998, 75–87 | MR | Zbl

[14] A. Yu. Karlovich, Yu. I. Karlovich, “One sided invertibility of binomial functional operators with a shift in rearrangement-invariant spaces”, Integral Equations Operator Theory, 42:2 (2002), 201–228 | DOI | MR | Zbl

[15] A. Antonevich, Ju. Makowska, “On spectral properties of weighted shift operators generated by mappings with saddle points”, Complex Anal. Oper. Theory, 2:2 (2008), 215–240 | DOI | MR | Zbl

[16] A. B. Antonevich, Yu. Yakubovska, “Weighted translation operators generated by mappings with saddle points: a model class”, J. Math. Sci. (N. Y.), 164:4 (2010), 497–517 | DOI | MR | Zbl

[17] A. B. Antonevich, A. A. Akhmatova, “Spektralnye svoistva diskretnogo operatora vzveshennogo sdviga”, Tr. In-ta matem., 20:1 (2012), 14–21

[18] A. Akhmatova, A. Antonevich, “On operators generated by maps with separable dynamics”, Geometric methods in physics, XXXI workshop (Poland, June 24–30, 2012), Birkhäuser, Basel, 2013, 171–178 | Zbl

[19] Yu. D. Latushkin, A. M. Stepin, “Weighted translation operators and linear extensions of dynamical systems”, Russian Math. Surveys, 46:2 (1991), 95–165 | DOI | MR | Zbl

[20] A. B. Antonevich, “Coherent local hyperbolicity of a linear extension and the essential spectra of a weighted shift operator on a closed interval”, Funct. Anal. Appl., 39:1 (2005), 9–20 | DOI | DOI | MR | Zbl

[21] Yu. D. Latushkin, A. M. Stepin, “Weighted shift operator on a topological Markov chain”, Funct. Anal. Appl., 22:4 (1988), 330–331 | DOI | MR | Zbl

[22] A. B. Antonevich, V. I. Bakhtin, A. V. Lebedev, “On $t$-entropy and variational principle for the spectral radii of transfer and weighted shift operators”, Ergodic Theory Dynam. Systems, 31:4 (2011), 995–1042 | DOI | MR | Zbl

[23] G. J. Murphy, $C^*$-algebras and operator theory, Academic Press, Inc., Boston, MA, 1990, x+286 pp. | MR | Zbl

[24] E. V. Kisin, “Algebry, porozhdennye dinamicheskimi sistemami i vzveshennymi sdvigami”, Dokl. AN SSSR, 219:2 (1974), 70–71

[25] V. S. Rabinovich, S. Roch, B. Silbermann, Limit operators and their applications in operator theory, Oper. Theory Adv. Appl., 150, Birkhäuser Verlag, Basel, 2004, xvi+392 pp. | DOI | MR | Zbl

[26] V. Rabinovich, S. Roch, “Essential spectrum of difference operators on periodic metric spaces”, Funct. Anal. Appl., 43:2 (2009), 151–154 | DOI | DOI | MR | Zbl

[27] W. C. Ridge, “Approximate point spectrum of a weighted shift”, Trans. Amer. Math. Soc., 147:2 (1970), 349–356 | DOI | MR | Zbl

[28] M. V. Marton, “Suschestvennye spektry Fredgolma, Veilya i Braudera operatorov vzveshennogo sdviga”, Vestn. BGU. Ser. 1, 2003, no. 1, 61–66 | MR | Zbl

[29] M. Bichegkuev, Spektralnaya teoriya raznostnykh i differentsialnykh operatorov i vyrozhdennye beskonechno differentsiruemye polugruppy operatorov, Dis. $\dots$ dokt. fiz.-matem. nauk, Voronezhskii gos. un-t., Voronezh, 2011, 32 pp.