Characterizations of Disjointness preserving operators on vector-valued function spaces

Jeang, Jyh-Shyang; Lin, Ying-Fen

doi:10.1090/s0002-9939-07-09086-7

Cited by 7 publications

(3 citation statements)

References 6 publications

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“…As a consequence, compactness, weak compactness and complete continuity are equivalent for disjointness preserving operators on continuous functions. However, we note that this equivalences do not hold in the case of vector-valued functions as shown in [10].…”

Section: Introductionmentioning

confidence: 81%

The structure of compact disjointness preserving operators on continuous functions

Lin

Wong

2009

Mathematische Nachrichten

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The duals of C 0 (a, b) and C[a, b] with respect to disjointness preserving groups are characterized. A. Plessner's result (1929) about the translation group is extended. A Wiener-Young type theorem for disjointness preserving groups is obtained. It is assumed that the reader is familiar with the notions of a Banach lattice, positive operator and disjointness preserving operator. [AB], [LZ] and [MN] are good references for this material. We also assume familiarity with the theory of C 0-semigroups and groups. [Pa] and [Na] should provide all necessary information when needed. Throughout −∞ ≤ a < b ≤ +∞, X is either (a, b) or [a, b], where by [−∞, b] we mean (−∞, b], and by [a, +∞] we mean [a, +∞). Definition. A function f : X → C is called vanishing at infinity if ∀ε > 0 ∃K ⊂ X, K compact such that ∀x ∈ X \ K |f (x)| < ε. We denote the lattice of all continuous functions f : X → C vanishing at infinity by C 0 (X), the lattice of all bounded continuous functions f : X → C by C b (X), the lattice of all continuous functions f : X → C with compact support by C c (X), and the lattice of all regular complex-valued Borel measures on X with finite variation by M (X). Let {T (t)} t∈R be a C 0-group on a Banach space E. Definition. The group dual of E with respect to {T (t)} t∈R , denoted E and pronounced E-sun, is defined in the following way: E = { u * ∈ E * : lim t→0 T * (t)u * − u * = 0 }. [vN] is an excellent source of information about the semigroup and group duals of Banach spaces and related subjects. Definition. A group {T (t)} t∈R on a Banach lattice E is called disjointness preserving if ∀t ∈ R T (t) is a disjointness preserving operator. Let {T (t)} t∈R be a disjointness preserving C 0-group on C 0 (X). It follows from [Na, B-II.3.8] that ∃ϕ : R × X → X a continuous flow ∃q : R → C b (X) a

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Section: Introductionmentioning

confidence: 81%

The structure of compact disjointness preserving operators on continuous functions

Lin

Wong

2009

Mathematische Nachrichten

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“…This explains the interest and amount of work devoted to the characterization of separating or biseparating operators. See, e.g., [1,2,3,4,5,7,8,16,19,27].…”

Section: Introductionmentioning

confidence: 99%

Biseparating maps on generalized Lipschitz spaces

Leung

2010

Studia Math.

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Let X, Y be complete metric spaces and E, F be Banach spaces. A bijective linear operator from a space of E-valued functions on X to a space of F -valued functions on Y is said to be biseparating if f and g are disjoint if and only if T f and T g are disjoint. We introduce the class of generalized Lipschitz spaces, which includes as special cases the classes of Lipschitz, little Lipschitz and uniformly continuous functions. Linear biseparating maps between generalized Lipschitz spaces are characterized as weighted composition operators, i.e., of the form T f (y) = Sy(f (h −1 (y)) for a family of vector space isomorphisms Sy : E → F and a homeomorphism h : X → Y . We also investigate the continuity of T and related questions. Here the functions involved (as well as the metric spaces X and Y ) may be unbounded. Also, the arguments do not require the use of compactification of the spaces X and Y .2000 Mathematics Subject Classification. 46E15; 47B38; 47B33.

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“…A similar result for operators T : C 0 (X) → C 0 (Y ), X, Y locally compact, was obtained in [16], as well as characterizations of compactness and weak compactness of T . These results were extended to the vector-valued case in [13]. In a parallel direction, many authors have studied algebraic homomorphisms between spaces of scalarvalued differentiable functions [4,9,10].…”

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confidence: 97%

Compact and weakly compact disjointness preserving operators on spaces of differentiable functions

Leung

Wang

2012

Trans. Amer. Math. Soc.

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A pair of functions defined on a set X with values in a vector space E is said to be disjoint if at least one of the functions takes the value 0 at every point in X. An operator acting between vector-valued function spaces is disjointness preserving if it maps disjoint functions to disjoint functions. We characterize compact and weakly compact disjointness preserving operators between spaces of Banach space-valued differentiable functions.2010 Mathematics Subject Classification. Primary 46E40, 46E50, 47B33, 47B38. Key words and phrases. Disjointness preserving operators, spaces of vector-valued differentiable functions, compact and weakly compact operators.

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Characterizations of Disjointness preserving operators on vector-valued function spaces

Abstract: Abstract. We characterize compact and completely continuous disjointness preserving linear operators on vector-valued continuous functions as follows: a disjointness preserving operator T : C 0 (X, E) → C 0 (Y, F ) is compact (resp. completely continuous) if and only if

Cited by 7 publications

References 6 publications

The structure of compact disjointness preserving operators on continuous functions

The structure of compact disjointness preserving operators on continuous functions

Biseparating maps on generalized Lipschitz spaces

Compact and weakly compact disjointness preserving operators on spaces of differentiable functions

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