Derivable mappings at unit operator on nest algebras

Zhu, Jiang; Xiong, Changping

doi:10.1016/j.laa.2006.12.002

Cited by 49 publications

(19 citation statements)

References 14 publications

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“…It follows from the main theorem in [14] that A 11 is an inner derivation on algN . Thus there exists an operator A ∈ B(H ) such that…”

Section: All-derivable Points In 2 × 2 Upper Triangular Operator Matrmentioning

confidence: 99%

“…We know from the main theorem in [14] that I M is an all-derivable point of algN M for the strong operator topology. By imitating the proof of Theorem 2.2, we may get that …”

Section: All-derivable Points In Continuous Nest Algebrasmentioning

confidence: 99%

“…Li, Pan and Xu [11] prove that every derivable mapping ϕ at 0 with ϕ(I ) = 0 on CSL algebras is a derivation. Zhu and Xiong in [13][14][15] showed that (1) every norm-continuous generalized derivable mapping at 0 on finite CSL algebras is a generalized derivation;…”

Section: Introductionmentioning

confidence: 99%

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All-derivable points in continuous nest algebras

Zhu¹,

Xiong²

2008

Journal of Mathematical Analysis and Applications

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Let A be an operator algebra on a Hilbert space. We say that an element G ∈ A is an all-derivable point of A for the strong operator topology if every strong operator topology continuous derivable linear mapping ϕ at G (i.e. ϕ(ST ) = ϕ(S)T + Sϕ(T ) for any S, T ∈ algN with ST = G) is a derivation. Let N be a continuous nest on a complex and separable Hilbert space H . We show in this paper that every orthogonal projection operator P (M) (0 = M ∈ N ) is an all-derivable point of algN for the strong operator topology.

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“…It follows from the main theorem in [14] that A 11 is an inner derivation on algN . Thus there exists an operator A ∈ B(H ) such that…”

Section: All-derivable Points In 2 × 2 Upper Triangular Operator Matrmentioning

confidence: 99%

“…We know from the main theorem in [14] that I M is an all-derivable point of algN M for the strong operator topology. By imitating the proof of Theorem 2.2, we may get that …”

Section: All-derivable Points In Continuous Nest Algebrasmentioning

confidence: 99%

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All-derivable points in continuous nest algebras

Zhu¹,

Xiong²

2008

Journal of Mathematical Analysis and Applications

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“…But so far all known results were obtained under some additional assumptions on nests or the spaces (Ref. [1,8,15]). The problem of finding all-derivable points of any nest algebras on any Banach spaces was attacked by [12] recently.…”

Section: A Linear Map δ : a → A Where A Is An Algebra Is A Derivatimentioning

confidence: 99%

All-derivable subsets for nest algebras on Banach spaces

Zhang¹,

Hou²,

Qi³

2014

imf

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Let N be a nest on a complex Banach space X and let AlgN be the associated nest algebra. We say that a subset S ⊂ AlgN is an all-derivable subset of AlgN if every linear map δ from AlgN into itself derivable on S (i.e. δ satisfies that, for each Z ∈ S, δ(A)B + Aδ(B) = δ(Z) for any A, B ∈ AlgN with AB = Z) is a derivation. In this paper, we show that S is an all-derivable subset of AlgN if span{ran(Z) : Z ∈ S} is dense in X or ∩{ker Z : Z ∈ S} = {0}. Mathematics Subject Classification: 47B47, 47L35

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“…In [23], Zhao and Zhu characterized Jordan derivable mappings at 0 and I on triangular algebras. (see [3,4,9,10,11,13,14,15,16,17,19,23,24,25,26] and the references therein for more related results). Motivated by these facts, we introduce the concept of the mappings (m, n)-derivable at some point which unifies the above definitions.…”

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confidence: 99%

On (m,n)-Derivations of Some Algebras

Shen

Guo

2014

Demonstratio Mathematica

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Abstract. Let A be a unital algebra, δ be a linear mapping from A into itself and m, n be fixed integers. We call δ an (m, n)-derivable mapping at Z, if mδpABq`nδpBAq " mδpAqB`mAδpBq`nδpBqA`nBδpAq for all A, B P A with AB " Z. In this paper, (m, n)-derivable mappings at 0 (resp. IA ' 0, I) on generalized matrix algebras are characterized. We also study (m, n)-derivable mappings at 0 on CSL algebras. We reveal the relationship between this kind of mappings with Lie derivations, Jordan derivations and derivations. IntroductionLet R be a unital ring and A be a unital R-algebra. Let δ be a linear mapping from A into itself. We call δ a derivation if δpABq " δpAqBÀ δpBq for all A, B P A. We call δ a Jordan derivation if δpAB`BAq " δpAqB`AδpBq`δpBqA`BδpAq for all A, B P A. δ is called a Lie derivation if δprA, Bsq " rδpAq, Bs`rA, δpBqs for all A, B P A, where rA, Bs " AB´BA. The questions of characterizing Jordan derivations and Lie derivations have received considerable attention from several authors, who revealed the relationship between Jordan derivations, derivations as well as Lie derivations (for example, [1,5,6,8,12] and the references therein).Let m, n be fixed integers. In [21], Vukman defined a new type of Jordan derivations, named pm, nq-Jordan derivation, that is, an additive mapping η from a ring R into itself such that pm`nqηpA 2 q " 2mηpAqA`2nAηpAq for every A P R. He proved that each pm, nq-Jordan derivation of a prime ring is a derivation. Motivated by this, we define a new type of derivations, named pm, nq-derivation. An pm, nq-derivation is a linear mapping δ from A into itself such that mδpABq`nδpBAq " mδpAqB`mAδpBq`nδpBqA`nBδpAq 2010 Mathematics Subject Classification: Primary 47L35; Secondary 16W25. Key words and phrases: CSL algebra, derivation, generalized matrix algebra, pm, nqderivation. for all A, B P A. Obviously, every p1, 1q-derivation is a Jordan derivation, each p1,´1q-derivation is a Lie derivation, p1, 0q-derivations and (0, 1)-derivations are derivations.Recently, there have been a number of papers on the study of conditions under which derivations on algebras can be completely determined by their action on some subsets of elements. Let δ be a linear mapping from A into itself and Z be in A. δ is called derivable at Z, if δpABq " δpAqB`AδpBq for all A, B P A with AB " Z; δ is called Jordan derivable at Z, if δpAB`BAq " δpAqB`AδpBq`δpBqA`BδpAq for all A, B P A with AB " Z; δ is called Lie derivable at Z, if δprA, Bsq " rδpAq, Bs`rA, δpBqs for all A, B P A with AB " Z. It is natural and interesting to ask whether or not a linear mapping is a derivation (Jordan derivation, or Lie derivation) if it is derivable (Jordan derivable, Lie derivable) only at one given point. An and Hou [2] investigated derivable mappings at 0, P, and I on triangular rings, where P is some fixed non-trivial idempotent. Let X be a Banach space, Lu and Jing [16] studied Lie derivable mappings at 0 and P on BpXq, where P is a fixed nontrivial idempotent. In [23], Zhao and Zhu characterized Jordan derivable mapping...

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Derivable mappings at unit operator on nest algebras

Cited by 49 publications

References 14 publications

All-derivable points in continuous nest algebras

All-derivable points in continuous nest algebras

All-derivable subsets for nest algebras on Banach spaces

On (m,n)-Derivations of Some Algebras

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