On the Mean Ergodic Theorem in Bicomplex Banach Modules
Tóm tắt
In this paper the mean ergodic theorem in bicomplex Banach modules is studied. Under appropriate conditions of boundness for the iterates compositions of a bicomplex linear and bounded operator on a bicomplex Banach module, and of weak compactness of the average sequence on the idempotent components, analogous to that of the classical case on Banach spaces, strong convergence of the mean sequence is achieved. Also, a result on ergodicity is given for bounded bicomplex strongly continuous semigroups in bicomplex Banach modules.
Tài liệu tham khảo
Alpay, D., Luna-Elizarrarás, M.E., Shapiro, M., Struppa, D.C.: Basics of Functional Analysis with Bicomplex Scalars, and Bicomplex Schur Analysis. Springer, Cham (2014)
Bajguz, W., Kwaśniewski, A.K.: On hyperbolic quasinumbers and Chebyshev-like functions. Rep. Math. Phys. 43(3), 367–376 (1999)
Banach, S.: Theory of Linear Operations. North-Holland, Amsterdam (1987) [Original in Polish: Theorié des Operations Linéaires. PWN-Państwowe Wydawnictwo Naukowe, Warsaw (1979)]
Bernstein, S., Legatiuk, D.: Brownian motion, martingales and Itô formula in Clifford algebras. Adv. Appl. Clifford Algebras 32(27), 18 (2022)
Birkhoff, G.D.: Proof of the ergodic theorem. Proc. Natl. Acad. Sci. U.S.A. 17, 656–660 (1931)
Bocchieri, P., Loinger, A.: Ergodic theorem in quantum mechanics. Phys. Rev. 111, 668–670 (1958)
Carathéodory, C.: Bemerkungen zum Ergodensatz von G. Birkhoff. S.-B. Math.-Nat. Kl. Bayer. Acad. Wiss. 208, 189–208 (1944)
Carathéodory, C.: Algebraic Theory of Measure and Integration. Chealsea Publishing Company, New York (1963) (Original in German: Mass Und Integral Und Ihre Algebraisierung. Birkhäuser Verlag, Basel, 1956)
Charak, K.S., Kumar, R., Rochon, D.: Infinite dimensional bicomplex spectral decomposition theorem. Adv. Appl. Clifford Algebras 23, 593–605 (2013)
Cohen, L.W.: On the mean ergodic theorem. Ann. Math. 41, 505–509 (1940)
Colombo, F., Sabatini, I., Struppa, D.: Bicomplex holomorphic functional calculus. Math. Nachr. 287(10), 1093–1105 (2014)
Engel, K.-J., Nagel, R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000)
Eriksson, S.-L., Kaarakka, T.: Hyperbolic harmonic functions and hyperbolic Brownian motion. Adv. Appl. Clifford Algebras 30(72), 13 (2020)
Ghiloni, R., Recupero, V.: Semigroups over real alternative \(*\)-algebras: generation theorems and spherical sectorial operators. Trans. Am. Math. Soc. 368(4), 2645–2678 (2016)
Hille, E., Phillips, R. S.: Functional Analysis and Semi-Groups. American Mathematical Society, Colloquium Publications, vol. 31. AMS, Providence (1957)
Jacobson, N.: Basic Algebra I. W. H. Freeman and Company, San Francisco (1974)
Jordan, P., von Neumann, J., Wigner, E.: On an algebraic generalization of the quantum mechanical formalism. Ann. Math. 35, 29–64 (1934)
Klein, M.I.: The ergodic theorem in quantum statistical mechanics. Phys. Rev. 87, 111–115 (1952)
Koumantos, P.N.: Order topology in Minkowski space and applications. Rep. Math. Phys. 87(2), 209–223 (2021)
Koumantos, P.N., Pavlakos, P.K.: Mean ergodic theorems in Jordan Banach weak algebras. J. Math. Syst. Sci. 3(3), 146–149 (2013)
Krengel, U.: Ergodic Theorems. Walter de Gruyter, Berlin (1985)
Kumar, R., Saini, H.: Topological bicomplex modules. Adv. Appl. Clifford Algebras 26, 1249–1270 (2016)
Kumar, R., Singh, K.: Bicomplex linear operators on bicomplex Hilbert spaces and Littlewood’s subordination theorem. Adv. Appl. Clifford Algebras 25, 591–610 (2015)
Kumar, R., Kumar, R., Rochon, D.: The fundamental theorems in the framework of bicomplex topological modules (2011). arXiv:1109.3424v1 [math.FA]
Kumar, R., Singh, K., Saini, H., Kumar, S.: Bicomplex weighted Hardy spaces and bicomplex \(C^{*}\)-algebras. Adv. Appl. Clifford Algebras 26, 217–235 (2016)
Kwaśniewski, A.K., Czech, R.: On quasi-number algebras. Rep. Math. Phys. 31(3), 341–351 (1992)
Luna-Elizarrarás, M.E., Perez-Regalado, C.O., Shapiro, M.: On linear functionals and Hahn–Banach theorems for hyperbolic and bicomplex modules. Adv. Appl. Clifford Algebras 24, 1105–1129 (2014)
Luna-Elizarrarás, M.E., Shapiro, M., Struppa, D.C., Vajiac, A.: Bicomplex Holomorphic Functions: The Algebra, Geometry and Analysis of Bicomplex Numbers. Birkhäuser, Springer International Publishing Switzerland, Cham (2015)
Moyal, J.E.: Mean ergodic theorems in quantum mechanics. J. Math. Phys. 10, 506–509 (1969)
Price, G.B.: An Introduction to Multicomplex Spaces and Functions. Marcel Dekker Inc, New York (1991)
Riesz, F.: Some mean ergodic theorems. J. Lond. Math. Soc. 13, 274–278 (1938)
Rochon, D., Tremblay, S.: Bicomplex quantum mechanics: I. The generalized Schrödinger equation. Adv. Appl. Clifford Algebras 14(2), 231–248 (2004)
Rochon, D., Tremblay, S.: Bicomplex quantum mechanics: II. The Hilbert space. Appl. Clifford Algebras 16, 135–157 (2006)
Saini, H., Sharma, A., Kumar, R.: Some fundamental theorems of functional analysis with bicomplex and hyperbolic scalars. Adv. Appl. Clifford Algebras 30(66), 23 (2020)
Schrader, R.: Markov structures and Clifford algebras. J. Funct. Anal. 18, 369–413 (1975)
Segal, I.: Postulates for general quantum mechanics. Ann. Math. 48, 930–948 (1947)
Segre, C.: Le rappresentazioni reali delle forme complesse e gli enti iperalgebrici. Math. Ann. 40(3), 413–467 (1892)
von Neumann, J.: Proof of the quasi-ergodic hypothesis. Proc. Natl. Acad. Sci. U.S.A. 18, 70–82 (1932)
Wreszinski, W.F.: Irreversibility, the time arrow and a dynamical proof of the second law of thermodynamics. Quantum Stud. Math. Found. 7, 125–136 (2020)
Yosida, K.: Mean ergodic theorem in Banach spaces. Proc. Imp. Acad. Jpn. 14, 292–294 (1938)
Yosida, K.: Functional Analysis, 6th edn. Springer, Berlin (1980)
Yosida, K., Kakutani, S.: Operator-theoretical treatment of Markoff’s processes and mean ergodic theorem. Ann. Math. 42, 188–228 (1941)