A survey of weighted substitution operators and generalizations of BanachStone theorem <s> continuity of θ is taken with respect to strong operator topology on B(E,F)). <s> Abstract Let X and Y be vector lattices and let T : X → Y be a disjointness preserving linear operator, i.e., | Tx 1 |∧| Tx 2 |=0 if | x 1 |∧| x 2 |=0; x 1 , x 2 ϵ X . Necessary and sufficient conditions are obtained under which T can be written as a multiplication, i.e., Tx(·)=e(·)x(τ(·)), where τ is a continuous mapping from a topological space on which Y can be represented as a vector lattice of extended real valued continuous functions into a topological space on which X can be similarly represented. If X and Y are normed lattices and T is continuous, these conditions are satisfied and hence T can be represented as such a multiplication. With the author's permission, several remarks made by C.B. Huijsmans and B. de Pagter are incorporated in the text. <s> BIB001 </s> A survey of weighted substitution operators and generalizations of BanachStone theorem <s> continuity of θ is taken with respect to strong operator topology on B(E,F)). <s> LetC(X,E) andC(Y,F) denote the spaces of continuous functions on the Tihonov spacesX andY, taking values in the Banach spacesE andF, respectively. A linear mapH:C(X,E)→C(Y,F) isseparating iff(x)g(x)=0 for allx inX impliesHf(y)Hg(y)=0 for ally inY. Some automatic continuity properties and Banach-Stone type theorems (i.e., asserting that isometries must be of a certain form) for separating mapsH between spaces of real- and complex-valued functions have already been developed. The extension of such results to spaces of vector-valued functions is the general subject of this paper. We prove in Theorem 4.1, for example, for compactX andY, that a linear isometryH betweenC(X,E) andC(Y,F) is a “Banach-Stone” map if and only ifH is “biseparating (i.e,H andH −1 are separating). The Banach-Stone theorems of Jerison and Lau for vector-valued functions are then deduced in Corollaries 4.3 and 4.4 for the cases whenE andF or their topological duals, respectively, are strictly convex. <s> BIB002 </s> A survey of weighted substitution operators and generalizations of BanachStone theorem <s> continuity of θ is taken with respect to strong operator topology on B(E,F)). <s> We construct an invertible disjointness preserving operator T on a normed lattice such that T−1 is not disjointness preserving. Two elements x1, x2 in an (Archimedean) vector lattice X are said to be disjoint (in symbols: x1 ⊥ x2) if |x1| ∧ |x2| = 0. Recall that a (linear) operator T : X → Y between vector lattices is said to be disjointness preserving if T sends any two disjoint elements in X to disjoint elements in Y, that is, Tx1 ⊥ Tx2 in Y whenever x1 ⊥ x2 in X . Assume now that a disjointness preserving operator T : X → Y is bijective, i.e., one-to-one and onto, so that T−1 : Y → X exists. It was conjectured by the first named author (see, for example, the problem section in [HL, page 143]) that T−1 is also disjointness preserving. The purpose of this note is to give a counterexample to this conjecture. Moreover, we will construct a bijective disjointness preserving automorphism T on a normed lattice G such that T−1 is not disjointness preserving. We precede our construction by several comments on some recent developments regarding the above conjecture. The two most important classes of vector lattices are the Banach lattices and the Dedekind complete vector lattices. The vector lattice G that we will construct is in neither of these classes. It cannot be a Banach lattice in view of the fact that, as has been recently shown by Huijsmans and de Pagter [HP] and independently by Koldunov [K], the above conjecture has an affirmative solution for Banach lattices. More precisely, they proved the following theorem. Theorem 1 ([HP, Theorem 2.1], [K, Theorem 3.6]). Let T be a one-to-one disjointness preserving operator from a relatively uniformly complete vector lattice X into a normed lattice Y . Then T is a d-isomorphism, that is, x1 ⊥ x2 in X if and only if Tx1 ⊥ Tx2 in Y . In particular, if TX = Y then T−1 : Y → X is a disjointness preserving operator. Since each Banach lattice is relatively uniformly complete, the previous theorem includes the case of Banach lattices. Here we should also cite the paper by K. Jarosz [J], which preceded [HP] and [K] and in which a special case of Theorem 1 was Received by the editors November 7, 1996. 1991 Mathematics Subject Classification. Primary 47B60. <s> BIB003
Let S 1 be a nonzero closed subspace of E. Then S 1 is said to be an M-summand of E, if there exists a closed proper subspace S 2 of E such that If E is a reflexive Banach space which does not have any nontrivial direct M-summand, then it has been shown by Cambern [4] that every linear surjective isometry from C 0 (X,E) to C 0 (Y ,E) can be represented as a weighted substitution operator. The problem is still unresolved in case of nonreflexive Banach spaces. Recall that an operator T from CV 0 (X,E) to CV 0 (X,E) is said to be disjointness pre- . In lattice language, T is disjointness preserving if f ⊥g implies that T f ⊥Tg. It turns out that in some cases, the class of disjointness preserving operators contains the class of surjective isometries, that is, every linear surjective isometry on C 0 (X,E) is a disjointness preserving operator for suitable Banach spaces. In general, if T is a linear surjective isometry such that T and T −1 are disjointness preserving, then T can be represented as a weighted substitution operator. We refer to BIB002 for further details. If E is strictly convex or E * is strictly convex, then every onto isometry is disjointness preserving. The disjointness preserving operators have been studied on the weighted spaces of continuous functions and Banach lattices. For detail, we refer to BIB001 BIB003  . If V = C + 0 (X), the set of all continuous positive functions vanishing at infinity on X, then CV 0 (X,E) is a seminormed linear space of continuous E-valued functions f on X such that v f vanishes at infinity for every v ∈ V . Here each seminorm is a weighted analog of sup norm. It has been shown that every continuous disjointness preserving operator T on CV 0 (X) is a weighted substitution operator, that is, T = W θ ϕ , where ϕ : X → X and θ : X → C such that ϕ is continuous on {x ∈ X : θ(x) = 0). A generalization of this result for CV 0 (X,E) has also been obtained. We present this generalization in the following theorem. Outline of the proof. Suppose T is disjointness preserving. Let t ∈ E and t * ∈ E * and let f ∈ CV 0 (X). Then f t ∈ CV 0 (X,E), where t is a disjointness preserving operator and hence there exist a self-map ϕ t * t , independent of t, and t * and a function