Quasi-automorphisms on \mathbf{B}(\mathcal{X})

Abstract

Let \mathcal{X} be a Banach space with \dim \mathcal{X}\ge 2 and \text{B}(\mathcal{X}) be the Banach algebra of all bounded linear operators on \mathcal{X}, \mathrm{\forall }A,B\in \text{B}(\mathcal{X}). Define a quasi-product by A\circ B=A+B-AB, and (\text{B}(\mathcal{X}),\circ) is a semigroup. In this paper, we mainly discuss the quasi-product automorphisms on \text{B}(\mathcal{X}). It is proved that a bijective map \phi on \text{B}(\mathcal{X}) is a quasi-product automorphism if and only if \phi is a ring automorphism.

Keywords

Quasi-product / quasi-isomorphism / ring isomorphism

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