**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.

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