A geometric approach to inequalities for the Hilbert-Schmidt norm

We show that if X and Y are two non-zero Hilbert-Schmidt operators, then for any lambda >= 0, cos(2) Theta(X,Y) <= 1/1 + lambda root cos Theta(vertical bar X*vertical bar,vertical bar Y*vertical bar) root cos Theta(vertical bar X*vertical bar,vertical bar Y*vertical bar) vertical bar < X, Y >vertical bar/parallel to X parallel to(2)parallel to Y parallel to(2) + lambda/1 + lambda cos Theta(vertical bar X*vertical bar,vertical bar Y*vertical bar) cos Theta(vertical bar X vertical bar,vertical bar Y vertical bar) <= cos Theta(vertical bar X*vertical bar,vertical bar Y*vertical bar) cos Theta(vertical bar X vertical bar,vertical bar Y vertical bar). Here Theta(A,B) denotes the angle between non-zero Hilbert-Schmidt operators A and B. This enables us to present some inequalities for the Hilbert-Schmidt norm. In particular, we prove that parallel to X parallel to(2)parallel to Y parallel to(2) <= root root 2 + 1/2 parallel to vertical bar X vertical bar + vertical bar Y vertical bar parallel to(2).