A survey of weighted substitution operators and generalizations of BanachStone theorem <s> Characterizations of isometries on some general spaces of continuous functions <s> The real-valued, bounded, continuous functions on a topological space form a Banach space which has been studied extensively. (See [8]2 for a review of this theory.) The present paper is concerned with extending the results of this theory to the space Bx of bounded continuous maps from the topological space X into the real Banach space B. It is natural to expect that if X and Y are compact3 spaces and Bx and BY are equivalent then X and Y are homeomorphic. Such a theorem was proved by Stone [9] when B is the space of real numbers. But an example (?3) shows that it is not true for arbitrary B. The result is obtained, however, with certain restrictions on B (Theorem 5.2). A strictly convex Banach space satisfies the restrictions on B, and for such a space an explicit formula is given for the relation between an equivalence of Bx and BY and a homeomorphism of X and Y (?6). The space BX, when B is strictly convex, is studied further, and this leads to a characterization of such spaces. The characterization is related to those of Arens and Kelley [1] and Myers [7] of the space of real-valued functions. <s> BIB001 </s> A survey of weighted substitution operators and generalizations of BanachStone theorem <s> Characterizations of isometries on some general spaces of continuous functions <s> Part 1 Introduction: definitions and historical background L(p)-spaces functional Banach spaces of functions locally convex function spaces. Part 2 Composition operators on L(p)-spaces: definitions, characterizations and examples invertible composition operators compact composition operators normality of composition operators weighted composition operators. Part 3 Composition operators on functional Banach spaces: general characterizations composition operators on spaces H(p)(D), H(p)(D(n)) and H(p)(D(n)) composition operators on H(p)(p(+)) composition operators on l(p)-spaces. Part 4 Composition operators on the weighted locally convex function spaces: introduction, characterization and classical results composition operators on the weighted locally convex function spaces invertible and compact composition operators on weighted function spaces. Part 5 Some applications of composition operators: isometries and composition operators ergodic theory and composition operators dynamical systems and composition operators homomorphisms and composition operators references symbol index subject index. <s> BIB002
Let X be a locally compact Hausdorff space. Let V be a set of positive real-valued uppersemicontinuous functions on X. Then V is said to be a system of weights on X if for v 1 ,v 2 ∈ V and α > 0, there exists v ∈ V such that αv i ≤ v for i = 1,2 and for every x ∈ X, there exists v ∈ V such that v(x) > 0. If V is the set of all positive constant functions on X or V is the set of all continuous positive functions vanishing at infinity or V is the set of positive scalar multiples of all characteristic functions, then V is a system of weights on X. If E is a Banach space and V is a system of weights on X, then we can define the spaces CV 0 (X,E) and CV b (X,E) in the following manner: The spaces CV 0 (X,E) and CV b (X,E) are linear spaces of continuous E-valued · v is a seminorm on either CV b (X,E) or CV 0 (X,E) and the family { · v : v ∈ V } of seminorms defines a Hausdorff locally convex topology on each of these spaces. The spaces CV 0 (X,E) and CV b (X,E) with the locally convex topologies induced by the above family of seminorms are called the weighted spaces of vector-valued continuous functions on X. These spaces are natural generalizations of If V is the system of constant weights, then CV 0 (X,E) = C 0 (X,E), the space of all continuous functions vanishing at infinity. If V is the set of all positive multiples of characteristic functions, then CV 0 (X,E) = CV b (X,E) = C 0 (X,E), the space of all E-valued continuous functions with compact supports. For further details about the weighted spaces of continuous functions and some operators on them, we refer to  BIB002   . At the place of the Banach space E, one can take any locally convex space and define the weighted spaces of vector-valued continuous functions which form a much larger class of locally convex spaces of continuous functions. In this section, we will report Banach-Stone-theorem-type results on some weighted spaces of vector-valued continuous functions. The first vector-valued generalization of Banach-Stone theorem was obtained by M. Jerison in 1950 in case of strictly convex Banach spaces E. Recall that a Banach space E is strictly convex if every vector of unit norm is an extreme point of the closed unit ball of Jerison BIB001 proved in the following theorem. 