Let T be a bounded linear operator on a complex Hilbert space \(\mathcal {H}\) . We present some necessary and sufficient conditions for T to be the generalized pencil \(P + \alpha Q +\beta PQ\) of a pair (P, Q) of projections at some point \((\alpha , \beta )\in \mathbb {C}^2\) . The range and kernel relations of the generalized pencil T are studied and comments on the additional properties of some special generalized pencil are given.
