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Noninvertibility preservers on Banach algebras

Bojan Kuzma (2006)

Czechoslovak Mathematical Journal

It is proved that a linear surjection Φ 𝒜 , which preserves noninvertibility between semisimple, unital, complex Banach algebras, is automatically injective.

Non-linear Jordan triple automorphisms of sets of self-adjoint matrices and operators

Lajos Molnár (2006)

Studia Mathematica

We consider the so-called Jordan triple automorphisms of some important sets of self-adjoint operators without the assumption of linearity. These transformations are bijective maps which satisfy the equality ϕ(ABA) = ϕ(A)ϕ(B)ϕ(A) on their domains. We determine the general forms of these maps (under the assumption of continuity) on the sets of all invertible positive operators, of all positive operators, and of all invertible self-adjoint operators.

Nonlinear mappings preserving at least one eigenvalue

Constantin Costara, Dušan Repovš (2010)

Studia Mathematica

We prove that if F is a Lipschitz map from the set of all complex n × n matrices into itself with F(0) = 0 such that given any x and y we know that F(x) - F(y) and x-y have at least one common eigenvalue, then either F ( x ) = u x u - 1 or F ( x ) = u x t u - 1 for all x, for some invertible n × n matrix u. We arrive at the same conclusion by supposing F to be of class ¹ on a domain in ℳₙ containing the null matrix, instead of Lipschitz. We also prove that if F is of class ¹ on a domain containing the null matrix satisfying F(0) = 0...

Nonlinear maps preserving Lie products on triangular algebras

Weiyan Yu (2016)

Special Matrices

In this paper we prove that every bijection preserving Lie products from a triangular algebra onto a normal triangular algebra is additive modulo centre. As an application, we described the form of bijections preserving Lie products on nest algebras and block upper triangular matrix algebras.

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