A geometric approach to Wigner‐type theorems

Let H be a complex Hilbert space and let P(H) be the associated projective space (the set of rank‐one projections). Suppose dimH⩾3. We prove the following Wigner‐type theorem: if H is finite dimensional, then every orthogonality preserving transformation of P(H) is induced by a unitary or anti‐unitary operator. This statement will be obtained as a consequence of the following result: every orthogonality preserving lineation of P(H) to itself is induced by a linear or conjugate‐linear isometry (H is not assumed to be finite‐dimensional). As an application, we describe (not necessarily injective) transformations of Grassmannians preserving some types of principal angles.