Abstract
In this article we give some generalized singular number inequalities for products and sums of τ-measurable operators. Some related arithmetic-geometric mean and Heinz mean inequalities for a generalized singular number of τ-measurable operators are proved.
Highlights
LetMn be the space of n ×n complex matrices
Using the notion of the generalized singular number studied by Fack and Kosaki [ ], we generalize inequalities ( . )-( . ) for τ -measurable operators associated with a semifinite von Neumann algebra M
From Lemma . in [ ] we see that the generalized singular number function t → μt(x) is decreasing right-continuous and μt(uxv) ≤ v u μt(x), t >, ( . )
Summary
The singular values of A, i.e., the eigenvalues of the operator |A|, enumerated in decreasing order, will be denoted by Sj(A), j = , , . The arithmetic-geometric mean inequality for singular values due to Bhatia and Kittaneh [ ] says that. N, for positive semidefinite matrices A, B ∈ Mn. On the other hand, Tao [ ] observed that if. It was pointed out in [ ] that inequalities ), Audenaert [ ] (see [ ]) gave a Heinz mean inequality for singular values, that is, if A, B ∈ Mn are positive semidefinite matrices and ≤ r ≤ , . Sj ArB –r + A –rBr ≤ Sj(A + B), j = , , .
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