The main result of this paper is a characterization of real finite-dimensional inner product spaces among the normed ones in terms of the orthogonality of linear operators acting on the spaces. More precisely, it is shown that the real finite-dimensional normed space \(X\) has its norm induced by an inner product if and only if for any two orthogonal linear operators \(A\) and \(B\) on \(X\), there exists a unit vector \(x\) in \(X\) such that \(\| A\|= \| Ax\|\) and \(Ax\) and \(Bx\) are orthogonal to each other. Here the orthogonality of \(A\) and \(B\) is defined as \(\| A\|\leq\| A+ \lambda B\|\) for every scalar \(\lambda\), and the orthogonality of vectors in \(X\) is defined similarly.
The necessity of this result was proved by R. Bhatia and P. Šemrl [ibid. 287, No. 1–3; 77–85 (1999; Zbl 0937.15023)]. The sufficiency is proved here first for the real two-dimensional normed spaces and then for the general case. The case for \(X= \ell^n_p\) with \(p\neq 2\) was verified by G. A. Leonov and K. R. Schneider [Lect. Notes Control Inf. Sci. 273, 241–254 (2002; Zbl 1102.93311)], refuting an earlier conjecture of Bhatia and Šemrl.