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Register a new userErgodic Theorems for Lower Probabilities
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We establish an Ergodic Theorem for lower probabilities, a generalization of standard probabilities widely used in applications. As a by-product, we provide a version for lower probabilities of the Strong Law of Large Numbers.
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In this paper we study the main properties of the Ces`aro means of bi-continuous semigroups, introduced and studied by K{u}hnemund in [24]. We also give some applications to Feller semigroups generated by second-order elliptic differential operators with unbounded coefficients in $C_b(R^N)$ and to evolution operators associated with nonautonomous second-order differential operators in $C_b(R^N)$ with time-periodic coefficients.
In this paper we generalize Kingmans sub-additive ergodic theorem to a large class of infinite countable discrete amenable group actions.
Generalized smooth functions are a possible formalization of the original historical approach followed by Cauchy, Poisson, Kirchhoff, Helmholtz, Kelvin, Heaviside, and Dirac to deal with generalized functions. They are set-theoretical functions defined on a natural non-Archimedean ring, and include Colombeau generalized functions (and hence also Schwartz distributions) as a particular case. One of their key property is the closure with respect to composition. We review the theory of generalized smooth functions and prove both the local and some global inverse function theorems.
Let $X$ and $Y$ be completely regular spaces and $E$ and $F$ be Hausdorff topological vector spaces. We call a linear map $T$ from a subspace of $C(X,E)$ into $C(Y,F)$ a emph{Banach-Stone map} if it has the form $Tf(y) = S_{y}(f(h(y))$ for a family of linear operators $S_{y} : E to F$, $y in Y$, and a function $h: Y to X$. In this paper, we consider maps having the property: cap^{k}_{i=1}Z(f_{i}) eqemptysetiffcap^{k}_{i=1}Z(Tf_{i}) eq emptyset, where $Z(f) = {f = 0}$. We characterize linear bijections with property (Z) between spaces of continuous functions, respectively, spaces of differentiable functions (including $C^{infty}$), as Banach-Stone maps. In particular, we confirm a conjecture of Ercan and Onal: Suppose that $X$ and $Y$ are realcompact spaces and $E$ and $F$ are Hausdorff topological vector lattices (respectively, $C^{*}$-algebras). Let $T: C(X,E) to C(Y,F)$ be a vector lattice isomorphism (respectively, *-algebra isomorphism) such that Z(f) eqemptysetiff Z(Tf) eqemptyset. Then $X$ is homeomorphic to $Y$ and $E$ is lattice isomorphic (respectively, $C^{*}$-isomorphic) to $F$. Some results concerning the continuity of $T$ are also obtained.
Versions of well known function theoretic operator theory results of Szego and Widom are established for the Neil algebra. The Neil algebra is the subalgebra of the algebra of bounded analytic functions on the unit disc consisting of those functions whose derivative vanishes at the origin.
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