On the norm of polynomials of two adjoint projections

Let f(X, Y) be a polynomial of two non-commuting variables and let P be an arbitrary nontrivial projection operator in Hilbert space. The class of all polynomials f(X, Y) for which {norm of matrix}f(P, P^*){norm of matrix} depends only on f and {norm of matrix}P{norm of matrix} are described. In the case when such a dependence exists the explicit formula is obtained. Some applications to singular integral operators are given.