A note on some equalities for frames in Hilbert spaces

Some equalities for frames involving the real parts of some complex numbers have been recently established in [P. Găvruta, On some identities and inequalities for frames in Hilbert spaces, J. Math. Anal. Appl., 321 (2006) 469–478]. In the current note, we generalize the equalities to a more general form which does not involve the real parts of the complex numbers.