REVERSE YOUNG-TYPE INEQUALITIES FOR MATRICES AND OPERATORS

We present some reverse Young-type inequalities for the Hilbert-Schmidt norm as well as any unitarily invariant norm. Furthermore, we give some inequalities dealing with operator means. More precisely, we show that if A, B is an element of B(H) are positive operators and r >= 0, A del B-r + 2r(A del B - A#B) <= A#B-r. We also prove that equality holds if and only if A = B. In addition, we establish several reverse Young-type inequalities involving trace, determinant and singular values. In particular, we show that if A and B are positive definite matrices and r >= 0, then tr ((1 + r)A - rB) <= tr vertical bar A(1+r)B(-r)vertical bar - r(root trA - root trB)(2).