A survey of weighted substitution operators and generalizations of BanachStone theorem <s> Introduction <s> A weighted endomorphism of an algebra is defined to be a linear operator which is equal to an endomorphism followed by a multiplier. Thus, if B is an algebra with unit, then T is a weighted endomorphism of B if there is an element u in B and an endomorphism S of B such that T: f u Sf. The problem considered in this note is to characterize compact weighted endomorphisms of C(X), the Banach algebra of continuous functions on a compact Hausdorff space X. If X is a compact Hausdorff space, then it is easy to see that every nonzero endomorphism of C(X) has the form Sf = f o qp for some continuous function 9p from X into X. Consequently, weighted endomorphisms of C(X) have the form f(x) -* u(x)f(qp(x)) for some u E C(X) and continuous function 9p from X into X. This map will be denoted by uCcp and its spectrum by a(uC,). In related papers, the spectra of weighted automorphisms of various algebras were considered by Kitover ([6] and [7]) and in a lattice theoretic setting by Arendt ([1] and [2]) and Schaeffer, Wolff and Arendt [8]. Results on weighted automorphisms and endomorphisms of the disc algebra may be found in [4] and [5]. We begin with some notation and terminology. If B, and B2 are Banach spaces, then a linear map T: B1 -B2 is compact if it maps bounded sets into sequentially compact sets. If X is a set and qp: X -* X, then we let qPn denote the nth iterate of qT, i.e. T0(x) = x and %, (x) = p(P(, 1(x)) for n > 0, x E X. Also, if qp: X -* X, then a point c in X is called a fixed point of 9p of order n if n is a positive integer, Tpn(c) = c and q)k(c) =-c, k = 1, . . . , n1. The principal result of this note will be the following theorem. <s> BIB001 </s> A survey of weighted substitution operators and generalizations of BanachStone theorem <s> Introduction <s> We characterize the compact and weakly compact weighted composition operators on the Banach space Cp(T, X) of those continuous functions from a completely regular Hausdorff space T into a Banach space X which have relatively compact range. The compact weighted endomorphisms of the Banach algebra C(T) of all real or complex valued continuous functions on a compact Hausdorff space T have been characterized by H. Kamowitz [1]. In particular, given u C C(T) and a continuous function 0: T T, Kamowitz has shown that the weighted composition operator uCO: C(T) -* C(T) (defined by uC+(f)(t) = u(t)f(q(t)) for f C C(T) and t C T) is compact if and only if each component of N(u) = {t C T: u(t) i 0} is contained in some open set V such that ObV is a constant function. For an arbitrary completely regular Hausdorff space T, however, if C(T) is replaced by the supremum normed algebra Cb(T) of all scalar valued bounded continuous functions on T, Kamowitz's condition does not guarantee that a weighted composition operator on Cb(T) is compact. Indeed, this can be readily seen through an obvious modification of the example constructed by Kamowitz [1, p. 519] to show that uC0 can fail to be compact even when X is constant on each component of N(u). A closer look at this same example also suggests a possible way aroiund the problem. In case T is compact, Kamowitz's result immediately yields that the following condition is at least necessary in order for uC0 to be compact: (*) ({t C T: lu(t)I > E}) is finite for each E > 0; the key to Kamowitz's example lies in the fact that this direct link between 0b and u does not hold. We will show that (*) is both necessary and sufficient for a weighted composltion operator on Cb(T) to be compact, whereby (*) is equiivalent to Kamowitz's condition in case T is compact. More importantly, we will demonstrate that (*) serves as well to characterize the weakly compact weighted composition operators on Cb(Y%1 Since our methods allow, we shall actually place our results in the broader context of weighted composition operators on spaces of vector valued continuous functions. Preliminaries. In what follows, we will use X to denote a (nontrivial) real or complex Banach space, while S and T will denote completely regular Hausdorff spaces. The set of all continuous functions from S into T will be designated by Received by the editors February 26, 1986. 1980 Mathematics Subject Classification (1985 Revisin). Primary 47B05, 47B38. (?)1987 American Mathematical Society 0002-9939/87 $1.00 + $.25 per page <s> BIB002 </s> A survey of weighted substitution operators and generalizations of BanachStone theorem <s> Introduction <s> A characterization of compact (and Al-weakly compact) weighted composition operators on real and complex Banach lattices which can be appropriately realized as function spaces is provided. The compact weighted composition operators on C b(X), all bounded continuous complex valued functions on X with the suprernum norm, have been characterized in the case of compact X by Kamowitz [4], and for completely regular X by Singh and Summers [6]. We recall that an operator T on C b(X) is called a weighted composition operator if there exists an r on Cb (X) and b a continuous map X from X to X so that for each f in C (X), we have Tf(x) = r(x)f(4)(x)). In this note we will characterize a class of compact weighted composition operators (and compact composition operators) on more general function spaces, which we will call Banach F-lattices. These include the Lp-spaces and the Banach lattices, both real and complex, with quasi-interior points (more generally, topological order partitions). We also demonstrate that these operators are equivalent to the M-weakly compact weighted composition operators. We will call a complex Banach space E a Banach F-lattice if there exists a real Banach lattice G, which can be identified with equivalence classes of real valued functions on a completely regular space X, and E = G + iG can be identified with equivalence classes of complex valued functions on X satisfying the following conditions: (i) If f is equivalent to g, then {x C X: f(x) = g(x)} is dense. (ii) Each continuous complex valued bounded function on X (or function with compact support in the case of a locally compact X ) represents an element in E. (iii) The vector space operations on E and the lattice operations on G correspond to the pointwise defined operations. Received by the editors September 26, 1988. Presented to the American Mathematical Society at the 94th Annual meeting on Janruar-y 8. 1988. 1980 l1athematics Subject Classification ( 1 985 Revi,sion). Primary 47B05, 47B38, 46B30. (?) 1990 American Mathematical Society 0002-9939/90 $1 00 + $.25 per page <s> BIB003 </s> A survey of weighted substitution operators and generalizations of BanachStone theorem <s> Introduction <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> BIB004 </s> A survey of weighted substitution operators and generalizations of BanachStone theorem <s> Introduction <s> In this paper, we prove that into isometries and disjointness preserving linear maps from $C_0(X)$ into $C_0(Y)$ are essentially weighted composition operators $Tf = h\cdot f\circ\varphi$ for some continuous map $\varphi$ and some continuous scalar-valued function $h$. <s> BIB005
Let X and Y be nonempty sets, let E be a Banach space, and let B(E) denote the Banach space of all bounded linear operators on E. Let F 1 (X,E) and F 2 (Y ,E) be linear spaces of E-valued functions on X and Y , respectively, with pointwise vector operations. If ϕ : Y → X is a map such that the composite function f • ϕ belongs to F 2 (Y ,E) for every f in F 1 (X,E), then the correspondence S ϕ taking f to f • ϕ is a linear transformation from F 1 (X,E) to F 2 (Y ,E) and it is called the substitution transformation (composition transformation) induced by ϕ. If θ : Y → C or θ : Y → B(E) is a map such that θ · f • ϕ belongs to F 2 (X,E) for every f in F 1 (X,E), then the mapping f → θ · f • ϕ is a linear transformation from F 1 (X,E) to F 2 (Y ,E) and it is denoted by W θ ϕ . The transformation W θ ϕ is called the weighted substitution transformation induced by the pair (θ,ϕ). In case the function spaces F 1 (X,E) and F 2 (Y ,E) have topologies compatible with the linear structures, and S ϕ and W θ ϕ are continuous, we call them substitution operator and weighted substitution operator, respectively. Theory of these operators had extensive developments during the last four decades or so and they have been studied on several function spaces of holomorphic functions, continuous functions, and measurable functions and their combinations. For details we refer to BIB003 10, BIB005 BIB001 BIB004 BIB002 . Let Y 1 be a subset of Y , ϕ : Y 1 → X and θ : Y 1 → C (or B(E)). For f ∈ F 1 (X,E), define θ · f • ϕ on Y as (θ · f • ϕ)(y) = θ(y) f (ϕ(y)) for y in Y 1 and (θ · f • ϕ)(y) = 0 for y in the complement of Y 1 . In case the correspondence f → θ · f • ϕ is continuous, it is called a generalized weighted substitution operator. We also denote it by W θ ϕ . The weighted substitution operators W θ ϕ are simple and natural type of operators on function spaces as they come out of two binary operations on functions, namely, Suppose C(X) is the space of all real-valued continuous functions on X. Then C(X) is a Banach lattice with f ∨g and f ∧g defined pointwise, that is, ( f ∨g)(x) = max{ f (x),g(x)} and f ∧ g = min{ f (x),g(x)}. A lattice homomorphism between two Banach lattices E and F is a linear mapping T : E → F such that T( f ∨ g) = T f ∨ Tg and T( f ∧ g) = T f ∧ Tg for every f ,g ∈ E. In the following theorem, we find all those linear functionals which are lattice homomorphisms from C(X) to R. Proof. If T : C(X) → R is a nonzero linear functional preserving the lattice structures, then by the Riesz representation theorem, there exists a Borel measure µ on X such that It follows from normality of X that the support of µ is a singleton set, say {x}. The converse is easy. It turns out in light of the above result that every nonzero lattice homomorphism between Banach lattices C(X) and C(Y ) is a weighted substitution operator. This is presented in the following theorem.  If T is a weighted substitution operator, then it is a lattice homomorphism. This completes the proof.