Norm of a linear combination of two operators on a Hilbert space

Let alpha, beta, gamma, delta be complex numbers such that gammadelta not equal 0. If A and B are bounded linear operators on the Hilbert space H such that gammaA + deltaB is right invertible then we study the operator norm of (alphaA + betaB) (gammaA + deltaB)(-1) using the angle 0 between two subspaces ran A and ran B or the angle psi = psi(A, B) between two operators A and B where cos psi(A, B) = sup{\<Af, Bf>\/(parallel toAfparallel to . parallel toBfparallel to); f is an element of H, Af not equal 0, Bf not equal 0}.