
TL;DR
This paper extends Ando's inequalities by exploring variations that compare differences of functions of matrices and their absolute differences using unitarily invariant norms.
Contribution
It introduces new variations of Ando's inequalities, broadening their applicability in matrix analysis and operator theory.
Findings
Derived new inequalities relating $f(B)-f(A)$ and $f(|B-A|)$
Extended the scope of Ando's inequalities to broader classes of functions
Provided bounds for unitarily invariant norms in matrix comparisons
Abstract
We give variations on Ando's result comparing and with respect to unitarily invariant norms on matrices.
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An extension of inequalities by Ando
Éric Ricard
Normandie Univ, UNICAEN, CNRS, Laboratoire de Mathématiques Nicolas Oresme, 14000 Caen, France
Abstract.
We give variations on Ando’s result comparing and with respect to unitarily invariant norms on matrices.
2010 Mathematics Subject Classification: 46L51; 47A30.
Key words: Norm inequalities, functional calculus
This note deals with norm matricial inequalities. Our starting point is the following inequality by Ando in [1]: if are positive matrices and is a unitarily invariant norm and is an operator monotone function on with , then
[TABLE]
The result was also obtained by Birman Koplienko and Solomyak in [3]. The inequality reverses if the reciprocal of is operator monotone, this holds for instance if is an increasing operator convex function with .
In [10], it is shown that for any ,
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This has been recently extended in [5] by Dinh, Ho, Le and Vo to any operator convex function with :
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The inequality is reversed if is non-negative operator monotone. It is naturally tempting to imagine that for any positive operator monotone function one has
[TABLE]
with reversed inequality for positive operator convex functions. The aim of this note is to show that such inequalities hold. To do so, we revisit Ando’s argument to see how it can be extended.
In the first section, we review basic facts on various comparisons of matrices before using them to deduce the main inequalities. We assume that the reader is familiar with matricial inequalities. We have chosen to stick with matricial inequalities but most of what is done here can be adapted to general semifinite von Neumann algebras.
1. Comparisons of matrices
We refer [2] for basic background on matricial inequalities. As usual, is the space of matrices of size over with its usual trace . We denote by its subset of positive semidefinite matrices.
Given , we denote by its singular values in decreasing order. We will frequently use that if and is a positive non-decreasing function then .
We recall classical orders beyond the usual one on selfadjoint matrices.
First for , we write if for all , .
If we set for the Ky Fan norms, Ky Fan’s principle (Theorem IV.2.2 in [2]) gives that if iff for any unitarily invariant norm. This is also equivalent to the existence of a completely positive map with and with see [8]. Using the polar decomposition, we obtain that iff there is a map , which is contractive for all unitarily invariant norms, so that . We won’t use it but we recall that if is a non-decreasing convex function and then if .
Finally, for and with , we write if for all , . Weyl’s monotonicity principle, Corollary III.2.3 in [2], gives that for with , . Thus, using the polar decomposition and diagonalization, it is easy to see that for and iff there are contractions so that and . Of course for , implies .
Now we gather some facts about these comparisons. They must be folklore but we give a proof for completeness. For , we write for \left[\begin{array}[]{cc}A&0\\ 0&B\end{array}\right]\in{\mathbb{M}}_{2n}.
Lemma 1.1**.**
If , then .
Proof.
Since is selfadjoint, it can be written as where and . It follows that for any , . Let and be the support projections of and , then
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The result then follows by Weyl’s monotony principle as is a contraction. ∎
Lemma 1.2**.**
Let and be non-decreasing functions and for such that , then .
Proof.
This is just the triangular inequality for the norms combined with the fact that . Indeed the are non-decreasing so it yields that for any . ∎
Lemma 1.3**.**
Let and be non-decreasing functions and so that for , then .
Proof.
By induction, it suffices to do it for . We have for some map which is contractive for all unitarily invariant norms. Let , since is non-decreasing there are positive reals so that . Hence . But ’s are contractions and
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This means that . Noticing that is a non-decreasing function of and , we get the conclusion by Lemma 1.2. ∎
Given a non-commutative polynomial in several variables
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we define as
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Lemma 1.4**.**
Let and be non-decreasing functions and such that for , then for any non-commutative polynomial of -variables, we have
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Proof.
This is just the combination of Lemmas 1.2 and 1.3. ∎
Remark 1.5**.**
One can extend the above lemmas in many ways. For instance, we can assume that we have a continuous sets of variables and replace sums by integrals against positive measures as long as the objects make sense.
2. Main inequalities
First we rewrite Ando’s proof from [1] (see also Section X in [2]). We fix and consider the function on , . For convenience, we set . These are the basic bricks for operator monotone functions.
Lemma 2.1**.**
Let , then for any .
Proof.
This is obvious if , we assume .
First we have the identity, f_{s}(A+D)-f_{s}(A)=s\big{(}(A+s)^{-1}-(A+D+s)^{-1}\big{)}. With , which is a contraction, we have . It follows that . But is unitarily equivalent to . As is operator monotone, we end up with . ∎
Lemma 2.2**.**
Let , then for all .
Proof.
Put and define as above. Then and by operator monotony of and Lemma 1.1 since ,
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Thanks to Lemma 2.1, we get
But as and commute and have disjoint supports, and are unitarily equivalent in and we can conclude. ∎
An operator monotone function with has an integral representation for some positive measure (that may charge [math]) such that . Thus Ando’s result, if follows from Lemma 2.2, as and the extension of Lemma 1.2 to integrals.
By Lemma 1.4, we directly get
Theorem 2.3**.**
Let and be operator monotone functions with for and be non-decreasing. Then for a non-commutative polynomial of variables and any and any matrices so that for , we have
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Remark 2.4**.**
One can see directly in the proof that one just need to assume . This can also be seen as applying Lemma 1.4 one more time, as .
The above theorem can be extended to more general objects other than polynomials and contains many particular cases. We give a few examples, assuming that are operator monotone functions with and are non-decreasing functions with . For any unitarily invariant norm and any , we have:
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Recall that if is operator convex on with then is operator monotone on by [7]. In particular, if is non-negative and , then for where is operator monotone. Thus a non-negative operator convex function on with has an integral representation:
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for some (positive) measure (that does not charge [math]) such that and some .
From those inequalities, one can also get results for operator convex functions, we give one example.
Theorem 2.5**.**
Let be a non-negative operator convex function on with and a non-decreasing function with , then for any , we have
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Proof.
We first prove it in the case where for some , by assumption . Note that for . It follows that for some . The function is operator monotone on with . By the triangular inequality for any , with :
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Let be the trace preserving conditional expectation onto the (commutative) algebra generated by , we have . If we denote by and be the support projections of and . As , and since , we get . Thus, and . Hence we arrive at |{\mathcal{E}}\big{(}\gamma((A+D)^{2}-A^{2})+\delta D\big{)}|\geqslant\gamma D^{2}+\delta|D|, from which for any
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Using inequality (4),
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As for any , we have s_{i}\big{(}h(|D|)\big{(}\gamma D^{2}+\delta|D|\big{)}\big{)}-s_{i}(h(|D|)g(|D|))=s_{i}(h(D))s_{i}(f(|D|)), we get
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The case of general follows by approximation. ∎
One can also adapt the arguments to get trace inequalities as in [4]. One gets for instance from (4) that if is an odd or even function non-decreasing on with and is operator monotone, then for all :
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The above arguments also give that if is an odd function non-decreasing on and is non-negative operator convex function with , then for all :
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We would like to remark that all of the above inequalities can be generalized to bounded operators with finite support on a semifinite von Neumann algebra (that one can assume to be a factor). One has to use the generalized -numbers of [6] instead of the singular values and symmetric function spaces instead of unitarily invariant norms see [9]. We leave other possible technical extensions to the interested readers.
We conclude by noticing that (1) does not hold for general concave functions. For instance, it is false for and the operator or the trace norms. Indeed, (1) for the operator norm would imply that is Lipschitz in that norm. By homogeneity and translation, this would imply that the absolute value is also Lipschitz in the operator norm on selfadjoint operators which is false.
Acknowledgments The author wishes to thanks Trung Hoa Dinh for exchanges that motivated this note.
The reference list from the paper itself. Each links out to its DOI / PubMed record.
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