scholarly journals Generalized Jordan N-Derivations of Unital Algebras with Idempotents

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Xinfeng Liang
Keyword(s):  
Commutative Ring ◽  
Linear Operators ◽  
Current Work ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Unital Algebra ◽  
Matrix Algebras ◽  
Mild Conditions ◽  
Nest Algebras ◽  
Triangular Algebras

Let A be a unital algebra with idempotent e over a 2-torsionfree unital commutative ring ℛ and S : A ⟶ A be an arbitrary generalized Jordan n-derivation associated with a Jordan n-derivation J . We show that, under mild conditions, every generalized Jordan n-derivation S : A ⟶ A is of the form S x = λ x + J x in the current work. As an application, we give a description of generalized Jordan derivations for the condition n = 2 on classical examples of unital algebras with idempotents: triangular algebras, matrix algebras, nest algebras, and algebras of all bounded linear operators, which generalize some known results.

Generators of Nest Algebras

1974 ◽  
Vol 26 (3) ◽  
pp. 565-575 ◽  
Author(s):  
W. E. Longstaff
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Nest Algebra ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Closed Algebra ◽  
Nest Algebras ◽  
Weakly Closed

A collection of subspaces of a Hilbert space is called a nest if it is totally ordered by inclusion. The set of all bounded linear operators leaving invariant each member of a given nest forms a weakly-closed algebra, called a nest algebra. Nest algebras were introduced by J. R. Ringrose in [9]. The present paper is concerned with generating nest algebras as weakly-closed algebras, and in particular with the following question which was first raised by H. Radjavi and P. Rosenthal in [8], viz: Is every nest algebra on a separable Hilbert space generated, as a weakly-closed algebra, by two operators? That the answer to this question is affirmative is proved by first reducing the problem using the main result of [8] and then by using a characterization of nests due to J. A. Erdos [2].

LIE-TYPE DERIVATIONS OF NEST ALGEBRAS ON BANACH SPACES AND RELATED TOPICS

2021 ◽  
pp. 1-40
Author(s):  
FENG WEI ◽  
YUHAO ZHANG
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Linear Mapping ◽  
Linear Operators ◽  
Future Research ◽  
Complex Field ◽  
Nest Algebra ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Nest Algebras

Abstract Let $\mathcal {X}$ be a Banach space over the complex field $\mathbb {C}$ and $\mathcal {B(X)}$ be the algebra of all bounded linear operators on $\mathcal {X}$ . Let $\mathcal {N}$ be a nontrivial nest on $\mathcal {X}$ , $\text {Alg}\mathcal {N}$ be the nest algebra associated with $\mathcal {N}$ , and $L\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$ be a linear mapping. Suppose that $p_n(x_1,x_2,\ldots ,x_n)$ is an $(n-1)\,$ th commutator defined by n indeterminates $x_1, x_2, \ldots , x_n$ . It is shown that L satisfies the rule $$ \begin{align*}L(p_n(A_1, A_2, \ldots, A_n))=\sum_{k=1}^{n}p_n(A_1, \ldots, A_{k-1}, L(A_k), A_{k+1}, \ldots, A_n) \end{align*} $$ for all $A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$ if and only if there exist a linear derivation $D\colon \text {Alg}\mathcal {N}\longrightarrow \mathcal {B(X)}$ and a linear mapping $H\colon \text {Alg}\mathcal {N}\longrightarrow \mathbb {C}I$ vanishing on each $(n-1)\,$ th commutator $p_n(A_1,A_2,\ldots , A_n)$ for all $A_1, A_2, \ldots , A_n\in \text {Alg}\mathcal {N}$ such that $L(A)=D(A)+H(A)$ for all $A\in \text {Alg}\mathcal {N}$ . We also propose some related topics for future research.

DRAZIN INVERSES IN JÖRGENS ALGEBRAS OF BOUNDED LINEAR OPERATORS

2008 ◽  
Vol 108 (1) ◽  
pp. 81-87
Author(s):  
Lisa A. Oberbroeckling
Keyword(s):  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear

Cline’s formula for g-Drazin inverses in a ring

Filomat ◽  
2019 ◽  
Vol 33 (8) ◽  
pp. 2249-2255
Author(s):  
Huanyin Chen ◽  
Marjan Abdolyousefi
Keyword(s):  
Operator Theory ◽  
Spectral Properties ◽  
Associative Ring ◽  
Linear Operators ◽  
Drazin Inverse ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Cline’S Formula

It is well known that for an associative ring R, if ab has g-Drazin inverse then ba has g-Drazin inverse. In this case, (ba)d = b((ab)d)2a. This formula is so-called Cline?s formula for g-Drazin inverse, which plays an elementary role in matrix and operator theory. In this paper, we generalize Cline?s formula to the wider case. In particular, as applications, we obtain new common spectral properties of bounded linear operators.

Nonlocal fractional semilinear differential inclusions with noninstantaneous impulses and of order α ∈ (1, 2)

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
JinRong Wang ◽  
Ahmed G. Ibrahim ◽  
Donal O’Regan ◽  
Adel A. Elmandouh
Keyword(s):  
Differential Inclusions ◽  
Mild Solutions ◽  
Multivalued Function ◽  
Linear Operators ◽  
Solution Set ◽  
Cosine Family ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Noninstantaneous Impulses

AbstractIn this paper, we establish the existence of mild solutions for nonlocal fractional semilinear differential inclusions with noninstantaneous impulses of order α ∈ (1,2) and generated by a cosine family of bounded linear operators. Moreover, we show the compactness of the solution set. We consider both the case when the values of the multivalued function are convex and nonconvex. Examples are given to illustrate the theory.

Convergence of Random Bounded Linear Operators in the Skorokhod Space

2018 ◽  
Author(s):  
Nelson Vadori ◽  
Anatoliy V. Swishchuk
Keyword(s):  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Skorokhod Space
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