Nonlinear Lie derivations of triangular algebras

In this paper we prove that every nonlinear Lie derivation of triangular algebras is the sum of an additive derivation and a map into its center sending commutators to zero.     

Higher derivations of triangular algebras and its generalizations

Let R be a commutative ring with identity, A and B be unital algebras over R and M be a unital (it A,it B)-bimodule. Let be the triangular algebra consisting of A, it Band M. Motivated by the work of Cheung [14] we mainly consider the question whether every higher derivation on a triangular algebra is an inner higher derivation. We also give some characterizations on (generalized-)Jordan (triple-)higher derivations of triangular algebras.     

Characterizations of Lie derivations of triangular algebras

Let T be a triangular algebra over a commutative ring R. In this paper, under some mild conditions on T, we prove that if δ:T→T is an R-linear map satisfying

δ([x,y])=[δ(x),y]+[x,δ(y)]     

Generalized Lie derivations on skew elements of prime algebras with involution

We characterize generalized Lie derivations on skew elements of prime algebras A with involution, provided that A does not satisfy polynomial identities of low degree. The analogous result for matrix algebras is also described.     

Invariant subspaces of operator Lie algebras and Lie algebras with compact adjoint action

It is proved that a Lie algebra of compact operators with a non-zero Volterra ideal is reducible (has a nontrivial invariant subspace). A number of other criteria of reducibility for collections of operators is obtained. The results are applied to the structure theory of Lie algebras of compact operators and normed Lie algebras with compact adjoint action.     

Generalized Jordan derivations on triangular matrix algebras

In this note, we prove that every generalized Jordan derivation from the algebra of all upper triangular matrices over a commutative ring with identity into its bimodule is the sum of a generalized derivation and an antiderivation.     

Jordan higher derivations on triangular algebras

In this paper, we show that any Jordan higher derivation on a triangular algebra is a higher derivation. This extends the main result in [13] to the case of higher derivations.     

Jordan derivations of triangular algebras

In this note, it is shown that every Jordan derivation of triangular algebras is a derivation.     

On generalized Lie derivations of Lie ideals of prime algebras

Let A be a prime algebra of characteristic not 2 with extended centroid C, let R be a noncentral Lie ideal of A and let B be the subalgebra of A generated by R. If f,d:R→A are linear maps satisfying that

     

Local derivations on certain CSL algebras

In this paper, it is shown that every norm continuous linear local derivation from an arbitrary CSL algebra whose lattice is generated by finitely many independent nests into any ultraweakly closed subalgebra which contains the algebra is an inner derivation, and that every norm continuous linear local derivation from an arbitrary CSL algebra whose lattice is completely distributive into any ultraweakly closed subalgebra which contains the algebra is a derivation.     

Additive maps on rings behaving like derivations at idempotent-product elements

For every ring R with the unit I containing a nontrivial idempotent P, we describe the additive maps δ from R into itself which behave like derivations, and show that derivations on such kinds of rings can be determined by the action on the elements A,B∈R with AB=0, AB=P and AB=I respectively. Those results of An and Hou [R. An, J. Hou, Characterizations of derivations on triangular rings: additive maps derivable at idempotents, Linear Algebra Appl. 431 (2009) 1070-1080], Bres?ar [M. Bres?ar, Characterizing homomorphisms, multipliers and derivations in rings with idempotents, Proc. Roy. Soc. Edinburgh. Sect. A. 137 (2007) 9-21] and Chebotar et al. [M.A. Chebotar, W.-F. Ke, P.-H. Lee, Maps characterized by action on zero products, Pacific J. Math. 216 (2) 2004 217-228] are improved.     

Characterizations of generalized derivations associated with hochschild 2-cocycles and higher derivations

Let be a unital Banach algebra and be a unital -bimodule. A bilinear mapping α : is called a Hochschild 2-cocycle if xα(y, z) ? α(xy, z) + α(x, yz) ? α(x, y)z = 0 for any . We show that if δ is a linear mapping from into satisfying δ(xy) = δ(x)y + xδ(y) + α(x, y) for any with xy = W, where is a left or right separating point of , then δ is a generalized Jordan derivation associated with a Hochschild 2-cocycle α. We also find the relation of higher derivations and generalized derivations associated with Hochschild 2-cocycles.     

Lengths of central linear generalized polynomials in matrix algebras

Let D be a division algebra finite-dimensional over its center C and let A=M_m(D), the m×m matrix ring over D. By the length of a linear generalized polynomial (GP) ?(X), we mean the least positive integer n such that ?(X) can be represented in the form for some . We denote by L(?)=n the length of ?. By a central linear GP for A we mean a nonzero linear GP with central values on A. In this paper we characterize all central linear GPs for A and determine the lengths of all central linear GPs for A.     

Generalized left derivations acting as homomorphisms and anti-homomorphisms on Lie ideal of rings

Let R be a prime ring with characteristic different from 2 and L be a Lie ideal of R. In this paper, we characterize generalized left derivation, which acts as a homomorphisms or an anti-homomorphisms on L.     

Jordan maps on triangular algebras

Let T be a triangular algebra and R′ be an arbitrary ring. Suppose that M:T→R^′ and M^∗:R^′→T are surjective maps such that

     

On images of locally finite derivations and E-derivations

Some cases of the LFED Conjecture, proposed by the second author [15], for certain integral domains are proved. In particular, the LFED Conjecture is completely established for the field of fractions $k(x)$ of the polynomial algebra $k[x]$, the formal power series algebra $k[[x]]$ and the Laurent formal power series algebra $k[[x]][x^{?1}]$, where $x=(x_1,x_2,\dots ,x_n)$ denotes n commutative free variables and k a field of characteristic zero. Furthermore, the relation between the LFED Conjecture and the Duistermaat–van der Kallen Theorem [3] is also discussed and emphasized.     

Characterizations of derivations of Banach space nest algebras: All-derivable points

Let N be a nest on a complex Banach space X with N∈N complemented in X whenever N_-=N, and let AlgN be the associated nest algebra. We say that an operator Z∈AlgN is an all-derivable point of AlgN if every linear map δ from AlgN into itself derivable at Z (i.e. δ(A)B+Aδ(B)=δ(Z) for any A,B∈A with AB=Z) is a derivation. In this paper, it is shown that if Z∈AlgN is an injective operator or an operator with dense range, or an idempotent operator with ran(Z)∈N, then Z is an all-derivable point of AlgN. Particularly, if N is a nest on a complex Hilbert space, then every idempotent operator with range in N, every injective operator as well as every operator with dense range in AlgN is an all-derivable point of the nest algebra AlgN.     

Determination of a spacewise dependent heat source

This paper investigates the inverse problem of determining a spacewise dependent heat source in the parabolic heat equation using the usual conditions of the direct problem and information from a supplementary temperature measurement at a given single instant of time. The spacewise dependent temperature measurement ensures that the inverse problem has a unique solution, but this solution is unstable, hence the problem is ill-posed. For this inverse problem, we propose an iterative algorithm based on a sequence of well-posed direct problems which are solved at each iteration step using the boundary element method (BEM). The instability is overcome by stopping the iterations at the first iteration for which the discrepancy principle is satisfied. Numerical results are presented for various typical benchmark test examples which have the input measured data perturbed by increasing amounts of random noise.