Singular value and unitarily invariant norm inequalities for sums and products of operators

In this note, we mainly investigate singular value and unitarily invariant norm inequalities for sums and products of operators. First, we present singular value inequality for the quantity AX+YB : let A, B, X and Y is an element of B(H) such that both A and B are positive operators. Then sj(((AX+YB) circle plus 0) <= s(j) (((K + M) circle plus (L-1 + N)), for j=1,2, ..., where K=1/2A+1/2A(1/2)vertical bar X*vertical bar(2)A(1/2) L-1=1/2B+1/2B(1/2)|Y|B-2(1/2) M=1/2|B-1/2(X+Y)*A(1/2) and N=1/2|A(1/2)(X+Y)B-1/2|. In addition, based on the above singular value inequality, we establish a unitarily invariant norm inequality for concave functions. These results generalize inequalities obtained by Audeh directly. Finally, we present another more general singular value inequality for Sigma(m)(i=1) Ai*Xi*Bi s(j) (Sigma(m)(i=1)A(i)*X-i*Bi circle plus 0) <= s(j) ((Sigma(m)(i=1)A(i)*f(i)(2)(|X-i|)A(i) circle plus (Sigma Bi-m(i=1)*gi2(|Xi*|)Bi)) j=1,2, where Ai Xi is an element of B(H) such that Ai B{i} (i=1,2, ... ,m are compact operators and (i=1,2, ... ,m are 2m nonnegative continuous functions on [0,+infinity) (i=1,2, ...mi=1,2,... ,m for t is an element of[0,+infinity).