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Multiplicative Lie triple derivations on standard operator algebras

Bilal Ahmad Wani (2021)

Communications in Mathematics

Let 𝒳 be a Banach space of dimension n > 1 and 𝔄 ⊂ ℬ ( 𝒳 ) be a standard operator algebra. In the present paper it is shown that if a mapping d : 𝔄 → 𝔄 (not necessarily linear) satisfies d ( [ [ U , V ] , W ] ) = [ [ d ( U ) , V ] , W ] + [ [ U , d ( V ) ] , W ] + [ [ U , V ] , d ( W ) ] for all U , V , W ∈ 𝔄 , then d = ψ + τ , where ψ is an additive derivation of 𝔄 and τ : 𝔄 → 𝔽 I vanishes at second commutator [ [ U , V ] , W ] for all U , V , W ∈ 𝔄 . Moreover, if d is linear and satisfies the above relation, then there exists an operator S ∈ 𝔄 and a linear mapping τ from 𝔄 into 𝔽 I satisfying τ ( [ [ U , V ] , W ] ) = 0 for all U , V , W ∈ 𝔄 , such that d ( U ) = S U - U S + τ ( U ) for all U ∈ 𝔄 .

Multiplicative maps that are close to an automorphism on algebras of linear transformations

L. W. Marcoux, H. Radjavi, A. R. Sourour (2013)

Studia Mathematica

Let be a complex, separable Hilbert space of finite or infinite dimension, and let ℬ() be the algebra of all bounded operators on . It is shown that if φ: ℬ() → ℬ() is a multiplicative map(not assumed linear) and if φ is sufficiently close to a linear automorphism of ℬ() in some uniform sense, then it is actually an automorphism; as such, there is an invertible operator S in ℬ() such that φ ( A ) = S - 1 A S for all A in ℬ(). When is finite-dimensional, similar results are obtained with the mere assumption that there...

Multiplicative structure of de Branges's spaces.

Benjamin A. Lotto, Donald Sarason (1991)

Revista Matemática Iberoamericana

L. de Branges has originated a viewpoint one of whose repercussions has been the detailed analysis of certain Hilbert spaces of holomorphic functions contained within the Hardy space H2 of the unit disk (...).

Multiplicatively and non-symmetric multiplicatively norm-preserving maps

Maliheh Hosseini, Fereshteh Sady (2010)

Open Mathematics

Let A and B be Banach function algebras on compact Hausdorff spaces X and Y and let ‖.‖X and ‖.‖Y denote the supremum norms on X and Y, respectively. We first establish a result concerning a surjective map T between particular subsets of the uniform closures of A and B, preserving multiplicatively the norm, i.e. ‖Tf Tg‖Y = ‖fg‖X, for certain elements f and g in the domain. Then we show that if α ∈ ℂ 0 and T: A → B is a surjective, not necessarily linear, map satisfying ‖fg + α‖X = ‖Tf Tg + α‖Y,...

Multiplicity results for sign-changing solutions of an operator equation in ordered Banach space

Xian Xu, Bingjin Wang (2009)

Mathematica Bohemica

In this paper, we prove some multiplicity results for sign-changing solutions of an operator equation in an ordered Banach space. The methods to show the main results of the paper are to associate a fixed point index with a strict upper or lower solution. The results can be applied to a wide variety of boundary value problems to obtain multiplicity results for sign-changing solutions.

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