Abstract
The well-known arithmetic-geometric mean inequality for singular values, due to Bhatia and Kittaneh, is one of the most important singular value inequalities for compact operators. The purpose of this study is to give new singular value inequalities for compact operators and prove that these inequalities are equivalent to arithmetic-geometric mean inequality, the way by which several future studies could be done.
Highlights
Fundamental PrinciplesLet B ( H ) indicate the set of all bounded linear operators on a complex separable Hilbert space H, and let K ( H ) indicate the two-sided ideal of compact operators in B ( H )
In this study, we will present several new inequalities, and prove that they are equivalent to arithmetic-geometric mean inequality.The following are the proved inequalities in this study: Let A, C and D be operators in K ( H ) where A ≥ 0, C and D arbitrary operators
The purpose of this study is to give new singular value inequalities for compact operators and prove that these inequalities are equivalent to arithmetic-geometric mean inequality, the way by which several future studies could be done
Summary
Let B ( H ) indicate the set of all bounded linear operators on a complex separable Hilbert space H, and let K ( H ) indicate the two-sided ideal of compact operators in B ( H ). If T ∈ K ( H ) , the singular values of T, denoted ( ) by s1 (T ) , s2 (T ) , are the eigenvalues of the positive operator T = T ∗T 1 2 ordered as s1 (T ) ≥ s2 (T ) ≥ and repeated according to multiplicity. It is well ( ) known that = s j (T ) s= j T ∗ s j ( T ). Zhan has proved in [6] that if A, B ∈ K ( H )
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