# Submajorization inequalities of $$\tau$$ τ -measurable operators for concave and convex functions

Submajorization inequalities of $$\tau$$ τ -measurable operators for concave and convex... Let $$M$$ M be a von Neumann algebra with a normal faithful semifinite trace $$\tau$$ τ . Let $$x_1 ,\ldots ,x_n$$ x 1 , … , x n be $$n\, \tau$$ n τ -measurable positive operators with respect to $$( {M,\tau })$$ ( M , τ ) , and let $$z_1 ,\ldots ,z_n$$ z 1 , … , z n be $$n$$ n expansive operators in $$M.$$ M . We prove that for a concave function $$f:\left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right) ,\, f(\sum \nolimits _{{k = 1}}^{n} {z_{k}^{*} x_{k} z_{k} } )$$ f : 0 , ∞ → 0 , ∞ , f ( ∑ k = 1 n z k ∗ x k z k ) is submajorised by $$\sum \nolimits _{{k = 1}}^{n} {z_{k}^{*} f(x_{k} )z_{k} }$$ ∑ k = 1 n z k ∗ f ( x k ) z k and the reverse submajorisation holds if $$f$$ f is a positive convex function with $$f( 0)=0.$$ f ( 0 ) = 0 . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Positivity Springer Journals

# Submajorization inequalities of $$\tau$$ τ -measurable operators for concave and convex functions

Positivity, Volume 19 (2) – Jul 11, 2014
5 pages

/lp/springer_journal/submajorization-inequalities-of-tau-measurable-operators-for-concave-N97EedntWk
Publisher
Springer Journals
Subject
Mathematics; Fourier Analysis; Operator Theory; Potential Theory; Calculus of Variations and Optimal Control; Optimization; Econometrics
ISSN
1385-1292
eISSN
1572-9281
D.O.I.
10.1007/s11117-014-0300-x
Publisher site
See Article on Publisher Site

### Abstract

Let $$M$$ M be a von Neumann algebra with a normal faithful semifinite trace $$\tau$$ τ . Let $$x_1 ,\ldots ,x_n$$ x 1 , … , x n be $$n\, \tau$$ n τ -measurable positive operators with respect to $$( {M,\tau })$$ ( M , τ ) , and let $$z_1 ,\ldots ,z_n$$ z 1 , … , z n be $$n$$ n expansive operators in $$M.$$ M . We prove that for a concave function $$f:\left[ 0,\infty \right) \rightarrow \left[ 0,\infty \right) ,\, f(\sum \nolimits _{{k = 1}}^{n} {z_{k}^{*} x_{k} z_{k} } )$$ f : 0 , ∞ → 0 , ∞ , f ( ∑ k = 1 n z k ∗ x k z k ) is submajorised by $$\sum \nolimits _{{k = 1}}^{n} {z_{k}^{*} f(x_{k} )z_{k} }$$ ∑ k = 1 n z k ∗ f ( x k ) z k and the reverse submajorisation holds if $$f$$ f is a positive convex function with $$f( 0)=0.$$ f ( 0 ) = 0 .

### Journal

PositivitySpringer Journals

Published: Jul 11, 2014

### References

• Eigenvalue inequalities for convex and log-convex functions
Aujla, JS; Bourin, J-C
• Weak majorization inequalities and convex functions
Aujla, JS; Silva, FC

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