Mathematica Bohemica, Vol. 142, No. 2, pp. 113-124, 2017
Weighted Frobenius-Perron operators and their spectra
Mohammad Reza Jabbarzadeh, Rana Hajipouri
Received December 15, 2015. First published November 22, 2016.
Abstract: First, some classic properties of a weighted Frobenius-Perron operator $\mathcal{P}_\varphi^u$ on $L^1(\Sigma)$ as a predual of weighted Koopman operator $W=uU_\varphi$ on $L^\infty(\Sigma)$ will be investigated using the language of the conditional expectation operator. Also, we determine the spectrum of $\mathcal{P}_\varphi^u$ under certain conditions.
References: [1] K. P. S. Bhaskara Rao, M. Bhaskara Rao: Theory of Charges: A Study of Finitely Additive Measures. Pure and Applied Mathematics 109, Academic Press, London (1983). MR 0751777 | Zbl 0516.28001 [2] J. T. Campbell, J. E. Jamison: On some classes of weighted composition operators. Glasg. Math. J. 32 (1990), 87-94 corrigendum on pages 261-263. DOI 10.1017/S0017089500009095 | MR 1045089 | Zbl 0705.47027 [3] J. Ding: A closed range theorem for the Frobenius-Perron operator and its application to the spectral analysis. J. Math. Anal. Appl. 184 (1994), 156-167. DOI 10.1006/jmaa.1994.1191 | MR 1275951 | Zbl 0804.47032 [4] J. Ding: The Frobenius-Perron operator as a product of two operators. Appl. Math. Lett. 9 (1996), 63-65. DOI 10.1016/0893-9659(96)00033-X | MR 1386001 | Zbl 0857.47016 [5] J. Ding: The point spectrum of Frobenius-Perron and Koopman operators. Proc. Am. Math. Soc. 126 (1998), 1355-1361. DOI 10.1090/S0002-9939-98-04188-4 | MR 1443148 | Zbl 0892.47010 [6] J. Ding, Q. Du, T. Y. Li: The spectral analysis of Frobenius-Perron operators. J. Math. Anal. Appl. 184 (1994), 285-301. DOI 10.1006/jmaa.1994.1200 | MR 1278389 | Zbl 0830.47022 [7] J. Ding, W. E. Hornor: A new approach to Frobenius-Perron operators. J. Math. Anal. Appl. 187 (1994), 1047-1058. DOI 10.1006/jmaa.1994.1405 | MR 1298836 | Zbl 0819.47043 [8] J. Ding, A. Zhou: On the spectrum of Frobenius-Perron operators. J. Math. Anal. Appl. 250 (2000), 610-620. DOI 10.1006/jmaa.2000.7003 | MR 1786085 | Zbl 0991.47014 [9] J. Ding, A. Zhou: Statistical Properties of Deterministic Systems. Tsinghua University Texts. Springer, Berlin; Tsinghua University Press, Beijing (2009). DOI 10.1007/978-3-540-85367-1 | MR 2518822 | Zbl 1171.37001 [10] M. R. Jabbarzadeh: Weighted Frobenius-Perron and Koopman operators. Bull. Iran. Math. Soc. 35 (2009), 85-96. MR 2642928 | Zbl 1203.47018 [11] M. R. Jabbarzadeh: A conditional expectation type operator on $L^p$ spaces. Oper. Matrices 4 (2010), 445-453. MR 2680958 | Zbl 1217.47068 [12] M. R. Jabbarzadeh, H. Emamalipour: Compact weighted Frobenius-Perron operators and their spectra. Bull. Iran. Math. Soc. 38 (2012), 817-826. MR 3028472 | Zbl 06283466 [13] M. R. Jabbarzadeh, M. Jafari Bakhshkandi: Stability constants for weighted composition operators. To appear in Bull. Belg. Math. Soc.-Simon Stevin. [14] M. M. Rao: Conditional Measures and Applications. Pure and Applied Mathematics (Boca Raton) 271, Chapman & Hall/CRC, Boca Raton (2005). MR 2149673 | Zbl 1079.60008 [15] K. Yosida: Functional Analysis. Classics in Mathematics. Vol. 123, Springer, Berlin (1995). DOI 10.1007/978-3-642-61859-8 | MR 1336382 | Zbl 0830.46001 [16] A. C. Zaanen: Integration. North-Holland, Amsterdam (1967). MR 0222234 | Zbl 0175.05002
Affiliations: Mohammad Reza Jabbarzadeh, Rana Hajipouri, Department of Pure Mathematics, Faculty of Mathematical Sciences, and Research Institute for Fundamental Sciences, University of Tabriz, 29 Bahman Blvd, P. O. Box 51664, Tabriz, 5166616471, Iran, e-mail: mjabbar@tabrizu.ac.ir, r.hajipouri@tabrizu.ac.ir
