Mappings on some reflexive algebras characterized by action on zero products or Jordan zero products

Abstract

Let L be a subspace lattice on a Banach space X and let δ : AlgL → B(X) be a linear mapping. If ∨{L ∈ L : L − L} = X or ∧{L − : L ∈ L, L − L} = (0), we show that the following three conditions are equivalent:whenever AB = 0, we obtain that δ is a generalized derivation and δ(I)A ∈ (AlgL) ′ for every A ∈ AlgL. We also prove that if ∨{L ∈ L : L − L} = X and ∧{L − : L ∈ L, L − L} = (0), then δ is a local generalized derivation if and only if δ is a generalized derivation.

Hyers-Ulam-Rassias stability of (m, n)-Jordan derivations

“…Let be a subspace lattice on a Hilbert space , define = E E : 0. For some properties of completely distributive subspace lattices and -subspace lattices, see [19,18]. A totally ordered subspace lattice is called a nest.…”

“…A totally ordered subspace lattice is called a nest. By [18,20], we know that if and satisfy one of the following conditions: = Alg and is a dual normal Banach -bimodule, where is a completely distributive subspace lattice on a Hilbert space ; then has a right or a left separating set with J ⊆ ( ). In [21], G. An and J. Li showed that if is a unital algebra and is a unital -bimodule with a right (left) separating set generated algebraically by idempotents in , then every ( ) m n , -Jordan derivation from a into is zero.…”

Characterizations of linear mappings through zero products or zero Jordan products

“…In [1,2,3,5,8,12,16,18,19,20,21,24,29,35], several authors consider the following conditions on a linear mapping δ from A into M: and they investigate whether these conditions characterize derivations or Jordan derivations.…”