Displaying similar documents to “Nonlinear * -Lie higher derivations of standard operator algebras”

Multiplicative Lie triple derivations on standard operator algebras

Bilal Ahmad Wani (2021)

Communications in Mathematics

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Let 𝒳 be a Banach space of dimension n > 1 and 𝔄 ( 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 .

Generalized Higher Derivations on Lie Ideals of Triangular Algebras

Mohammad Ashraf, Nazia Parveen, Bilal Ahmad Wani (2017)

Communications in Mathematics

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Let 𝔄 = 𝒜 be the triangular algebra consisting of unital algebras 𝒜 and over a commutative ring R with identity 1 and be a unital ( 𝒜 , ) -bimodule. An additive subgroup 𝔏 of 𝔄 is said to be a Lie ideal of 𝔄 if [ 𝔏 , 𝔄 ] 𝔏 . A non-central square closed Lie ideal 𝔏 of 𝔄 is known as an admissible Lie ideal. The main result of the present paper states that under certain restrictions on 𝔄 , every generalized Jordan triple higher derivation of 𝔏 into 𝔄 is a generalized higher derivation of 𝔏 into 𝔄 . ...

Annihilating and power-commuting generalized skew derivations on Lie ideals in prime rings

Vincenzo de Filippis (2016)

Czechoslovak Mathematical Journal

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Let R be a prime ring of characteristic different from 2 and 3, Q r its right Martindale quotient ring, C its extended centroid, L a non-central Lie ideal of R and n 1 a fixed positive integer. Let α be an automorphism of the ring R . An additive map D : R R is called an α -derivation (or a skew derivation) on R if D ( x y ) = D ( x ) y + α ( x ) D ( y ) for all x , y R . An additive mapping F : R R is called a generalized α -derivation (or a generalized skew derivation) on R if there exists a skew derivation D on R such that F ( x y ) = F ( x ) y + α ( x ) D ( y ) for all x , y R . We prove...

Centralizing traces and Lie-type isomorphisms on generalized matrix algebras: a new perspective

Xinfeng Liang, Feng Wei, Ajda Fošner (2019)

Czechoslovak Mathematical Journal

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Let be a commutative ring, 𝒢 be a generalized matrix algebra over with weakly loyal bimodule and 𝒵 ( 𝒢 ) be the center of 𝒢 . Suppose that 𝔮 : 𝒢 × 𝒢 𝒢 is an -bilinear mapping and that 𝔗 𝔮 : 𝒢 𝒢 is a trace of 𝔮 . The aim of this article is to describe the form of 𝔗 𝔮 satisfying the centralizing condition [ 𝔗 𝔮 ( x ) , x ] 𝒵 ( 𝒢 ) (and commuting condition [ 𝔗 𝔮 ( x ) , x ] = 0 ) for all x 𝒢 . More precisely, we will revisit the question of when the centralizing trace (and commuting trace) 𝔗 𝔮 has the so-called proper form from a new perspective. Using the aforementioned...

( φ , ϕ ) -derivations on semiprime rings and Banach algebras

Bilal Ahmad Wani (2021)

Communications in Mathematics

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Let be a semiprime ring with unity e and φ , ϕ be automorphisms of . In this paper it is shown that if satisfies 2 𝒟 ( x n ) = 𝒟 ( x n - 1 ) φ ( x ) + ϕ ( x n - 1 ) 𝒟 ( x ) + 𝒟 ( x ) φ ( x n - 1 ) + ϕ ( x ) 𝒟 ( x n - 1 ) for all x and some fixed integer n 2 , then 𝒟 is an ( φ , ϕ )-derivation. Moreover, this result makes it possible to prove that if admits an additive mappings 𝒟 , 𝒢 : satisfying the relations 2 𝒟 ( x n ) = 𝒟 ( x n - 1 ) φ ( x ) + ϕ ( x n - 1 ) 𝒢 ( x ) + 𝒢 ( x ) φ ( x n - 1 ) + ϕ ( x ) 𝒢 ( x n - 1 ) , 2 𝒢 ( x n ) = 𝒢 ( x n - 1 ) φ ( x ) + ϕ ( x n - 1 ) 𝒟 ( x ) + 𝒟 ( x ) φ ( x n - 1 ) + ϕ ( x ) 𝒟 ( x n - 1 ) , for all x and some fixed integer n 2 , then 𝒟 and 𝒢 are ( φ , ϕ )derivations under some torsion restriction. Finally, we apply these purely ring theoretic results to semi-simple Banach algebras. ...

Generalized reverse derivations and commutativity of prime rings

Shuliang Huang (2019)

Communications in Mathematics

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Let R be a prime ring with center Z ( R ) and I a nonzero right ideal of R . Suppose that R admits a generalized reverse derivation ( F , d ) such that d ( Z ( R ) ) 0 . In the present paper, we shall prove that if one of the following conditions holds: (i) F ( x y ) ± x y Z ( R ) , (ii) F ( [ x , y ] ) ± [ F ( x ) , y ] Z ( R ) , (iii) F ( [ x , y ] ) ± [ F ( x ) , F ( y ) ] Z ( R ) , (iv) F ( x y ) ± F ( x ) F ( y ) Z ( R ) , (v) [ F ( x ) , y ] ± [ x , F ( y ) ] Z ( R ) , (vi) F ( x ) y ± x F ( y ) Z ( R ) for all x , y I , then R is commutative.

On a Kirchhoff-Carrier equation with nonlinear terms containing a finite number of unknown values

Nguyen Vu Dzung, Le Thi Phuong Ngoc, Nguyen Huu Nhan, Nguyen Thanh Long (2024)

Mathematica Bohemica

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We consider problem (P) of Kirchhoff-Carrier type with nonlinear terms containing a finite number of unknown values u ( η 1 , t ) , , u ( η q , t ) with 0 η 1 < η 2 < < η q < 1 . By applying the linearization method together with the Faedo-Galerkin method and the weak compact method, we first prove the existence and uniqueness of a local weak solution of problem (P). Next, we consider a specific case ( P q ) of (P) in which the nonlinear term contains the sum S q [ u 2 ] ( t ) = q - 1 i = 1 q u 2 ( ( i - 1 ) q , t ) . Under suitable conditions, we prove that the solution of ( P q ) converges to the solution...

Σ s -products revisited

Reynaldo Rojas-Hernández (2015)

Commentationes Mathematicae Universitatis Carolinae

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We show that any Σ s -product of at most 𝔠 -many L Σ ( ω ) -spaces has the L Σ ( ω ) -property. This result generalizes some known results about L Σ ( ω ) -spaces. On the other hand, we prove that every Σ s -product of monotonically monolithic spaces is monotonically monolithic, and in a similar form, we show that every Σ s -product of Collins-Roscoe spaces has the Collins-Roscoe property. These results generalize some known results about the Collins-Roscoe spaces and answer some questions due to Tkachuk [Lifting the Collins-Roscoe...

Annihilators of skew derivations with Engel conditions on prime rings

Taylan Pehlivan, Emine Albas (2020)

Czechoslovak Mathematical Journal

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Let R be a noncommutative prime ring of characteristic different from 2, with its two-sided Martindale quotient ring Q , C the extended centroid of R and a R . Suppose that δ is a nonzero σ -derivation of R such that a [ δ ( x n ) , x n ] k = 0 for all x R , where σ is an automorphism of R , n and k are fixed positive integers. Then a = 0 .