Nonlinear commuting maps on ℒ (X)

ABSTRACT

Let X be a Banach space over the real or the complex field, and denote by $\mathcal{L}\left(X\right)$ the algebra of all bounded linear operators on it. We characterize maps $\varphi ,\psi :\mathcal{L}\left(X\right)\to \mathcal{L}\left(X\right)$ such that $\begin{array}{rl}(\varphi \left(T_1\right)+\psi \left(T_2\right))(T_1+T_2)& =(T_1+T_2)(\varphi \left(T_1\right)+\psi \left(T_2\right))\\ & (T_1,T_2\in \mathcal{L}\left(X\right)).\end{array}$

KEYWORDS:

• Commuting map
• nonlinear preserver problem
• commutant

AMS CLASSIFICATIONS:

• Primary 47B49
• Secondary 47B47

Disclosure statement

No potential conflict of interest was reported by the author.