Numerical Radius Preserving Linear Maps

International Mathematical Forum, 2014

Let B(X) be the algebra of all bounded linear operators on a complex Banach space X. Let x 0 is a nonzero fixed vector in X. We give the concrete form of every surjective map φ from B(X) into its self, such that the local spectral radius of φ(T)φ(S) + φ(R) at x 0 equals the local spectral radius of T S + R at x 0. We do not assume φ to be linear, or even additive.

Mathematical Sciences, 2012

Using the direct method based on the Hyers-Ulam-Rassias stability, we investigate and prove the Hyers-Ulam stability of Jordan homomorphisms in Jordan-Banach algebras for the functional equation

Mediterranean Journal of Mathematics, 2017

Let n ∈ N, A and B be Banach algebras and let B be a right A-module. We say that a linear mapping ϕ : A −→ B is a pseudo n-Jordan homomorphism if there exists an element w ∈ A such that ϕ(a n w) = ϕ(a) n • w, for every a ∈ A and n ≥ 2. In this paper, among other things, we show that under some conditions if a linear mapping ϕ is a (pseudo) n-Jordan homomorphism, then it is a (pseudo) (n + 1)-Jordan homomorphism. Additionally, we investigate automatic continuity of surjective pseudo n-Jordan homomorphisms under some conditions.

2015

The history of commutative algebra first appeared in 1890 by David Hilbert which was then followed by Banach spaces in 1924 since localization reduces many problems of geometric special case into commutative algebra problems of local ring. So far, many studies on preserver problems have been focusing on linear preserver problems (LPPs) especially LPPs in matrix theory. Also in consideration has been the characterization of all linear transformation on given linear space of matrices that leave certain functions, subsets and relations invariant. Clearly, we also have spectrum preserver problem or transmission. Kadison and Sourour have also shown that the derivation of local derivation of Von Neumann algebra R are continous linear maps if it coincides with some derivation at each point in the algebra over C . We employ the concept of 2-local automorphisms introduced by Serml that if we let A be an algebra, then the transformation is called a 2-local automorphism if for all x, y A the...

International Journal of Nonlinear Analysis and Applications, 2021

In this paper, we study some maps from a $C^*$-algebra into a Banach algebra or a Banach module. Under some conditions, by extending on unitization of Banach algebras, we prove that the maps defined into Banach algebras are homomorphisms and the others defined into Banach modules are derivations. Applications of our results in the context of $C^*$-algebras are also provided.

نشريه علوم رياضي و انفورماتيك, 2013

In this paper, we investigate the generalized Hyers-Ulam stability of Jordan homomorphisms in Jordan Banach algebras for the functional equation

Let A be a unital C *-algebra which has a faithful state. If φ : A → A is a unital linear map which is bijective and invert-ibility preserving or surjective and spectral radius preserving, then φ is a Jordan isomorphism. Also, we discuss other types of linear preserver maps on A.

Journal of Algebra, 1986

Linear Algebra and its Applications, 2017

Let L(X) be the algebra of all bounded linear operators on a complex Banach space X. We describe surjective linear maps φ on L(X) that satisfy r φ(T ) (x) = 0 =⇒ r T (x) = 0 for every x ∈ X and T ∈ L(X). We also describe surjective linear maps φ on L(X) that satisfy r T (x) = 0 =⇒ r φ(T ) (x) = 0 for every x ∈ X and T ∈ L(X). Furthermore, we characterize maps φ (not necessarily linear nor surjective) on L(X) which satisfy r φ(T )-φ(S) (x) = 0 if and only if r T -S (x) = 0 for every x ∈ X and T, S ∈ L(X).