Orthogonalities in linear spaces and difference operators

Abstract

Quite recently C. Alsina, P. Cruells and M. S. Tomas [2], motivated by F. Suzuki's property of isosceles trapezoids, have proposed the following orthogonality relation in a real normed linear space \( (X, \Vert \cdot \Vert) \): two vectors \( x,y \in X \) are T-orthogonal whenever¶\( \Vert z-x \Vert^2 + \Vert z-y \Vert^2 = \Vert z \Vert^2 + \Vert z-(x+y) \Vert^2 \)¶for every \( z \in X \). A natural question arises whether an analogue of T-orthogonality may be defined in any real linear space (without a norm structure). Our proposal reads as follows. Given a functional \( \varphi \) on a real linear space X we say that two vectors \( x,y \in X \) are \( \varphi \)-orthogonal (and write \( x\perp_{\varphi}y \)) provided that \( \Delta_{x,y}\varphi = 0 \) (\( \Delta_{h_1,h_2} \) stands here and in the sequel for the superposition \( \Delta_{h_1} \circ \Delta_{h_2} \) of the usual difference operators).¶We are looking for necessary and/or sufficient conditions upon the functional \( \varphi \) to generate a \( \varphi \)-orthogonality such that the pair \( X,\perp_{\varphi} \) forms an orthogonality space in the sense of J. Ratz (cf. [6]). Two new characterizations of inner product spaces as well as a generalization of some results obtained in [2] are presented.