Involutions of ℓ² and 𝑠 with unique fixed points

Abstract

Let σ ℓ 2 \sigma _{\ell ^2} and σ R ∞ \sigma _{\Bbb R^{\infty }} be the linear involutions of ℓ 2 \ell ^2 and R ∞ \mathbb {R}^\infty , respectively, given by the formula x → − x x\to -x . We prove that although ℓ 2 \ell ^2 and R ∞ \Bbb R^{\infty } are homeomorphic, σ ℓ 2 \sigma _{\ell ^2} is not topologically conjugate to σ R ∞ \sigma _{\Bbb R^{\infty }} . We proceed to examine the implications of this and give characterizations of the involutions that are conjugate to σ ℓ 2 \sigma _{\ell ^2} and to σ R ∞ \sigma _{\Bbb R^{\infty }} . We show that the linear involution x → − x x\to -x of a separable, infinite-dimensional Fréchet space E E is topologically conjugate to σ ℓ 2 \sigma _{\ell ^2} if and only if E E contains an infinite-dimensional Banach subspace and otherwise is linearly conjugate to σ R ∞ \sigma _{\Bbb R^{\infty }} .