  In this paper, based on the block operator technique and operator spectral
theory, the general explicit expressions for intertwining operators and direct
rotations of two orthogonal projections have been established. As a
consequence, it is an improvement of Kato's result (Perturbation Theory of
Linear operators, Springer-Verlag, Berlin/Heidelberg, 1996); J. Avron, R.
Seiler and B. Simon's Theorem 2.3 (The index of a pair of projections, J.
Funct. Anal. 120(1994) 220-237) and C. Davis, W.M. Kahan, (The rotation of
eigenvectors by a perturbation, III. SIAM J. Numer. Anal. 7(1970) 1-46).
