Abstract
By applying the theory of linear positive operators in a Banach space we derive spectral properties of certain composition operators in the Banach spaceA ∞(Ω) of holomorphic functions over some domain Ω⊂ℂ. Examples of such operators are provided by the so called generalized transfer matrices of classical one-dimensional lattice systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Viswanathan, K.S.: Statistical mechanics of a one-dimensional lattice gas with exponential polynomial interactions. Commun. Math. Phys.47, 131–141 (1976)
Mayer, D.H.: The transfer-matrix of a one-sided subshift of finite type with exponential interaction. Lett. Math. Phys.1, 335–343 (1976)
Mayer, D.H., Viswanathan, K.S.: On the ζ-function of a one-dimensional classical system of hardrods. Commun. Math. Phys.52, 175–189 (1977)
Mayer, D.H., Viswanathan, K.S.: Statistical mechanics of one-dimensional Ising and Potts models with exponential interactions. Physica89A, 97–112 (1977)
Mayer, D.H.: On composition operators on Banach spaces of holomorphic functions. Preprint, RWTH Aachen (1978); J. Funct. Anal. (in press)
Kac, M.: Mathematical mechanisms of phase transitions. In: Brandeis University Summer Institute in Theoretical Physics, Vol. 1, pp. 243–305. London: Gordon and Breach 1966
Perron, O.: Zur Theorie der Matrizen. Math. Ann.64, 248–263 (1907)
Frobenius, G.: Über Matrizen aus nicht negativen Elementen. S.-B. Preuss. Akad. Wiss. Berlin 456–477 (1912)
Jentzsch, R.: Über Integraloperatoren mit positivem Kern. Crelles J. Math.141, 235–244 (1912)
Krein, M.G., Rutman, M.A.: Linear operators leaving invariant a cone in a Banach space. Transl. Am. Math. Soc. Ser. 1,10, 199–325 (1950)
Krasnoselskii, M.A., Ladyzenskii, L.A.: Structure of a spectrum of positive non-homogeneous operators. Tr. Mosk. Mat. Obscestva3, (1954)
Krasnoselskii, M.A.: Positive solutions of operator equations, Chap. 2. Groningen, The Netherlands: Noordhoff 1964
Schäfen, H.H.: Topological vector spaces. Berlin, Heidelberg, New York: Springer 1971
See [11], p. 62, Theorem 2.2
See [11], Theorems 2.5, 2.10, 2.11, and 2.13
Dieudonne, J.: Foundations of modern analysis, p. 209. New York: Academic Press 1969
Herve, M.: Several complex variables, local theory, p. 83. Oxford: Oxford University Press 1963
Behnke, H., Thullen, P.: Theorie der Funktionen mehrerer komplexer Veränderlichen, Chap. V. 2. Aufl. Berlin, Heidelberg, New York: Springer 1970
Author information
Authors and Affiliations
Additional information
Communicated by H. Araki
Rights and permissions
About this article
Cite this article
Mayer, D.H. Spectral properties of certain composition operators arising in statistical mechanics. Commun.Math. Phys. 68, 1–8 (1979). https://doi.org/10.1007/BF01562537
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01562537