scholarly journals Maps Completely Preserving Involutions and Maps Completely Preserving Drazin Inverse

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Hongmei Yao ◽  
Baodong Zheng ◽  
Gang Hong
Keyword(s):  
Banach Spaces ◽  
Operator Algebras ◽  
Drazin Inverse ◽  
Complex Field ◽  
The Real ◽  
Infinite Dimensional ◽  
Standard Operator ◽  
Standard Operator Algebras ◽  
Preserver Problems

Let X and Y be infinite dimensional Banach spaces over the real or complex field 𝔽, and let 𝒜 and ℬ be standard operator algebras on X and Y, respectively. In this paper, the structures of surjective maps from 𝒜 onto ℬ that completely preserve involutions in both directions and that completely preserve Drazin inverse in both direction are determined, respectively. From the structures of these maps, it is shown that involutions and Drazin inverse are invariants of isomorphism in complete preserver problems.

A characterization of an isomorphism

2018 ◽  
Vol 11 (02) ◽  
pp. 1850022
Author(s):  
Ali Taghavi ◽  
Roja Hosseinzadeh ◽  
Efat Nasrollahi
Keyword(s):  
Banach Spaces ◽  
Operator Algebras ◽  
Standard Operator ◽  
Linear Maps ◽  
Standard Operator Algebras

Let [Formula: see text] and [Formula: see text] be some standard operator algebras on complex Banach spaces [Formula: see text] and [Formula: see text], respectively, and [Formula: see text] be a polynomial with no repeated roots and [Formula: see text], such that [Formula: see text]. We characterize the forms of surjective linear maps [Formula: see text] which preserve the nonzero products of operators that annihilated by [Formula: see text].

scholarly journals Maps preserving 2-idempotency of certain products of operators

Filomat ◽  
2017 ◽  
Vol 31 (12) ◽  
pp. 3909-3916
Author(s):  
Hossein Khodaiemehr ◽  
Fereshteh Sady
Keyword(s):  
Banach Spaces ◽  
Operator Algebras ◽  
Triple Product ◽  
Jordan Product ◽  
Standard Operator ◽  
Scalar Multiple ◽  
Standard Operator Algebras ◽  
Jordan Triple Product ◽  
Usual Product

Let A,B be standard operator algebras on complex Banach spaces X and Y of dimensions at least 3, respectively. In this paper we give the general form of a surjective (not assumed to be linear or unital) map ? : A ? B such that ?2 : M2(C)?A ? M2(C)?B defined by ?2((sij)2x2) = (?(sij))2x2 preserves nonzero idempotency of Jordan product of two operators in both directions. We also consider another specific kinds of products of operators, including usual product, Jordan semi-triple product and Jordan triple product. In either of these cases it turns out that ? is a scalar multiple of either an isomorphism or a conjugate isomorphism.

Mappings which Preserve Idempotents, Local Automorphisms, and Local Derivations

1993 ◽  
Vol 45 (3) ◽  
pp. 483-496 ◽  
Author(s):  
Matej Brešar ◽  
Peter Šemrl
Keyword(s):  
Local Ring ◽  
Operator Algebras ◽  
Banach Algebras ◽  
Continuous Linear ◽  
Matrix Algebras ◽  
Standard Operator ◽  
Weakly Continuous ◽  
Standard Operator Algebras ◽  
Jordan Homomorphisms

AbstractIt is proved that linear mappings of matrix algebras which preserve idempotents are Jordan homomorphisms. Applying this theorem we get some results concerning local derivations and local automorphisms. As an another application, the complete description of all weakly continuous linear surjective mappings on standard operator algebras which preserve projections is obtained. We also study local ring derivations on commutative semisimple Banach algebras.

scholarly journals On certain functional equations related to Jordan *-derivations in semiprime *-rings and standard operator algebras

2018 ◽  
Vol 27 (1) ◽  
pp. 1-17
Author(s):  
Mohammad Ashraf ◽  
Bilal Ahmad Wani
Keyword(s):  
Operator Algebras ◽  
Additive Mapping ◽  
Complex Hilbert Space ◽  
Linear Mapping ◽  
Linear Operators ◽  
Bounded Linear ◽  
Standard Operator ◽  
Semiprime Rings ◽  
Standard Operator Algebras ◽  
Torsion Free

Abstract The purpose of this paper is to investigate identities with Jordan *-derivations in semiprime *-rings. Let ℛ be a 2-torsion free semiprime *-ring. In this paper it has been shown that, if ℛ admits an additive mapping D : ℛ→ℛsatisfying either D(xyx) = D(xy)x*+ xyD(x) for all x,y ∈ ℛ, or D(xyx) = D(x)y*x*+ xD(yx) for all pairs x, y ∈ ℛ, then D is a *-derivation. Moreover this result makes it possible to prove that if ℛ satis es 2D(xn) = D(xn−1)x* + xn−1D(x) + D(x)(x*)n−1 + xD(xn−1) for all x ∈ ℛ and some xed integer n ≥ 2, then D is a Jordan *-derivation under some torsion restrictions. Finally, we apply these purely ring theoretic results to standard operator algebras 𝒜(ℋ). In particular, we prove that if ℋ be a real or complex Hilbert space, with dim(ℋ) > 1, admitting a linear mapping D : 𝒜(ℋ) → ℬ(ℋ) (where ℬ(ℋ) stands for the bounded linear operators) such that $$2D\left( {A^n } \right) = D\left( {A^{n - 1} } \right)A^* + A^{n - 1} D\left( A \right) + D\left( A \right)\left( {A^* } \right)^{n - 1} + AD\left( {A^{n - 1} } \right)$$ for all A∈𝒜(ℋ). Then D is of the form D(A) = AB−BA* for all A∈𝒜(ℋ) and some fixed B ∈ ℬ(ℋ), which means that D is Jordan *-derivation.

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