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 self-adjoint, 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 self-adjoint, B≥0, and ±A≤B, then
2s_{j}(A)≤s_{j}((B+A)⊕(B-A))
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 well-known arithmetic-geometric 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 Bhatia-Kittaneh, Kittaneh, Hirzallah-Kittaneh, Hirzallah, and Wang-Du. 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.
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