6th Summer Workshop on Operator Theory

9th - 13th July 2018 Kraków

Jacek Chmieliński

On operators reversing Birkhoff orthogonality

Let (X,∥·∥) be a normed space over the scalar field K and let ⊥B denote the Birkhoff orthogonality, i.e.,

x⊥B y ⇔ ∀λ ∈ K:∥x+λy∥ ≥ ∥x∥  (x,y∈X).

A linear operator T: X→ X reverses orthogonality (cf. [1]) iff

x⊥B y ⇒ Ty⊥B Tx,  x,y∈ X.

As opposed to the planar case, if dim X ≥ 3, the existence of such operators characterizes inner product spaces.

We consider also operators which \textit{approximately} reverse orthogonality. They may exist also in higher dimensional normed spaces which are not inner product ones.

The talk partially refers to a joint work with Paweł Wójcik.

  1. J. Chmieliński: Operators reversing orthogonality in normed spaces, Adv. Oper. Theory, 1 (2016), 8-14.
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