Boundedness, compactness and Schatten-class membership of weighted composition operators

The boundedness and compactness of weighted composition operators on the Hardy space ${\mathcal H}^2$ of the unit disc is analysed. Particular reference is made to the case when the self-map of the disc is an inner function. Schatten-class membership is also considered; as a result, stronger forms of the two main results of a recent paper of Gunatillake are derived. Finally, weighted composition operators on weighted Bergman spaces $\mathcal{A}^2 \alpha(\mathbb{D})$ are considered, and the results of Harper and Smith, linking their properties to those of Carleson embeddings, are extended to this situation.Comment: 12 page

Asymptotics of the Farey Fraction Spin Chain Free Energy at the Critical Point

We consider the Farey fraction spin chain in an external field $h$. Using ideas from dynamical systems and functional analysis, we show that the free energy $f$ in the vicinity of the second-order phase transition is given, exactly, by $$f \sim \frac t{\log t}-\frac1{2} \frac{h^2}t \quad \text{for} \quad h^2\ll t \ll 1 .$$ Here $t=\lambda_{G}\log(2)(1-\frac{\beta}{\beta_c})$ is a reduced temperature, so that the deviation from the critical point is scaled by the Lyapunov exponent of the Gauss map, $\lambda_G$. It follows that $\lambda_G$ determines the amplitude of both the specific heat and susceptibility singularities. To our knowledge, there is only one other microscopically defined interacting model for which the free energy near a phase transition is known as a function of two variables. Our results confirm what was found previously with a cluster approximation, and show that a clustering mechanism is in fact responsible for the transition. However, the results disagree in part with a renormalisation group treatment

Abel's Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions

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Langenbruch (Oldenburg) for providing a copy of [29].Bonet Solves, JA.; Domanski, P. (2015). Abel's Functional Equation and Eigenvalues of Composition Operators on Spaces of Real Analytic Functions. Integral Equations and Operator Theory. 81(4):455-482. https://doi.org/10.1007/s00020-014-2175-4S455482814Abel, N.H.: Determination d’une function au moyen d’une equation qui ne contient qu’une seule variable. In: Oeuvres Complètes, vol. II, pp. 246-248. Christiania (1881)Baker I.N.: Zusammensetzung ganzer Funktionen. Math. Z. 69, 121–163 (1958)Baker I.N.: Permutable power series and regular iteration. J. Aust. Math. Soc. 2, 265–294 (1961)Baker I.N.: Permutable entire functions. Math. Z. 79, 243–249 (1962)Baker I.N.: Fractional iteration near a fixpoint of multiplier 1. J. Aust. Math. Soc. 4, 143–148 (1964)Baker I.N.: Non-embeddable functions with a fixpoint of multiplier 1. Math. Z. 99, 337–384 (1967)Baker I.N.: On a class of nonembeddable entire functions. J. Ramanujan Math. 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