Maps preserving common zeros between subspaces of vector-valued continuous functions

Abstract

For metric spaces X and Y , normed spaces E and F, and certain subspaces A(X, E) and A(Y, F) of vector-valued continuous functions, we obtain a complete characterization of linear and bijective maps T : A(X, E) → A(Y, F) preserving common zeros, that is, maps satisfying the property. Moreover, we provide some examples of subspaces for which the automatic continuity of linear bijections having the property (P) is derived.

Kaplansky Theorem for completely regular spaces

“…We note that a biseparating mapping might not be nonvanishing preserving as shown in Example 1.2. The following lemma is motivated by the results in [6,17]. Proof.…”

Orthogonal forms and orthogonality preservers on real function algebras

“…Following a very usual technique (see, for example, [9,11,12,44,45]), we can define a continuous support map ϕ : Z 1 ∪ Z 2 → L 1 . More concretely, for each s ∈ Z 1 ∪ Z 2 , we write supp(δ s T ) for the set of all t ∈ L 1 such that for each open set U ⊆ L 1 with t ∈ U there exists f ∈ C r (L 1 ) with coz( f ) ⊆ U and δ s (T ( f )) = 0.…”

“…It is easy to see that Z 3 is closed. Following a very usual technique (see, for example, [8,9,30,16] and [17]), we can define a continuous support map ϕ :…”

Common zeros preserving maps on vector-valued function spaces and Banach modules

“…Here Z(·) denotes the zero set of a function. In the case where X, Y are metric spaces and E, F are normed spaces, a complete description of linear bijections T : A(X, E) → A(Y, F ) between certain subspaces of C(X, E) and C(Y, F ), satisfying the weaker condition Z(f ) ∩ Z(g) = ∅ ⇐⇒ Z(T f ) ∩ Z(T g) = ∅, for all f, g ∈ A(X, E), is given in [7] and then some extensions of the previous results are obtained. In [18], among other things, the authors considered maps preserving zero set containments, which dates back to [11].…”

“…A map T : A(X, E) → A(Y, F ) preserves common zeros if the pair T , T jointly preserves common zeros. In Section 3, we study such maps for certain subspaces of vector-valued continuous functions on Hausdorff spaces with values in a Hausdorff topological vector space and obtain generalizations of the results in [7]. In particular, for the spaces of Lipschitz functions we give an extension of [5,Theorem 6].…”