W. Audeh, K. Fuad, " Singular Value Inequalities for Compact Operators " , " ELSEVIER",Vol.437,No., Linear algebra and its applications, Amman, Jordan, 06/15/2012
Abstract:
A singular value inequality due to Bhatia and Kittaneh says that if A and B are compact operators on a complex separable Hilbert space such that A is selfadjoint, B≥0, and ±A≤B, then
s_{j}(A)≤s_{j}(B⊕B)
for j=1,2,... We give an equivalent inequality, which states that if A,B, and C are compact operators such that [
A B
B^{∗} C
]≥0, then
s_{j}(B)≤s_{j}(A⊕C)
for j=1,2,... Moreover, we give a sharper inequality and we prove that this inequality is equivalent to three equivalent inequalities considered by Tao. In particular, we show that if A and B are compact operators such that A is selfadjoint, B≥0, and ±A≤B, then
2s_{j}(A)≤s_{j}((B+A)⊕(BA))
for j=1,2,... Some applications of these results will be given.
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W. Audeh, " More Results on Singular Value Inequalities for Compact Normal Operators " , "Scientific Research Publishing",Vol.10,No., Advances in linear algebra and matrix theory, Amman, Jordan, 03/15/2013
Abstract:
The wellknown arithmeticgeometric mean inequality for singular values, due to Bhatia and Kittaneh says that if A and B are compact operators on a complex separable Hilbert space, then
2s_j (AB^* )≤s_j (A^* A+B^* B)
for j=1,2,... Hirzallah has proved that if A₁,A₂,A₃,and A₄ are compact operators, then
√2 s_j (A_1 A_2^*+A_3 A_4^* ^(1/2) )≤s_j ([■(A_1&A_3@A_2&A_4 )] )
for j=1,2,...We give inequality which is equivalent to and more general than the above inequalities, which states that if A_(i,),B_i,i=1,2,…,n are compact operators, then
2s_j (A_1 B_1^*+A_2 B_2^*+⋯+A_n B_n^* )≤s_j [■(A_1&A_2&⋯&A_n@0&0&⋯&0@⋮&⋮&⋮&⋮@0&0&0&0)^2+■(B_1&B_2&⋯&B_n@0&0&⋯&0@⋮&⋮&⋮&⋮@0&0&0&0)^2 ]
for j=1,2,...
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W. Audeh, " Singular Value Inequalities for Compact Normal Operators " , "Scientific Research Publishing",Vol.10,No., Advances in linear algebra and matrix theory, Amman, Jordan, 03/20/2013
Abstract:
We give singular value inequality to compact normal operators, which states that if A is compact normal operator on a complex separable Hilbert space, where A=A_1+iA_2 is the cartesian decomposition of A, then
1/√2 s_j (A_1+A_2)≤s_j (A)≤s_j (A_1 +A_2 )
for j=1,2,... Moreover, we give inequality which asserts that if A is compact normal operator, then
√2 s_j (A_1+A_2)≤s_j (A+iA^*)≤2s_j (A_1+A_2)
for j=1,2,... Several inequalities will be proved.
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W. Audeh, " More Commutator Inequalities for Hilbert Space Operators " , "Pushpa Publishing House",Vol.,No., International Journal of Functional Analysis, Operator Theory and Matrices, Allahabad, India, 06/12/2014
Abstract:
We present general singular value inequalities for nth order Audeh generalized commutator from them recent results for commutators due to BhatiaKittaneh, Kittaneh, HirzallahKittaneh, Hirzallah, and WangDu. are special cases. Several applications are given.
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Odeh, W, " Norm inequalities for finite Sums of positive operators " , "",Vol.,No., , Amman, Jordan, 04/16/2009
Abstract:
In this thesis we prove norm inequalities and singular value inequalities for finite sums of positive operators.

