An asymmetric Kadison's inequality
aa r X i v : . [ m a t h . F A ] M a r AN ASYMMETRIC KADISON’S INEQUALITY
JEAN-CHRISTOPHE BOURIN AND ´ERIC RICARD
Abstract.
Some inequalities for positive linear maps on matrix algebras aregiven, especially asymmetric extensions of Kadison’s inequality and severaloperator versions of Chebyshev’s inequality. We also discuss well-known resultsaround the matrix geometric mean and connect it with complex interpolation.
Introduction
This note lies in the scope of matricial inequalities. The main motivation ofthis theory is to extend some classical inequalities for reals to self-adjoint matrices.Of course, the non-commutativity of M n (the space of n × n complex matrices)enters into the game, making things much more complicated. The book [4] is avery good introduction to this subject. Many techniques have been developed,such as the theory of operator monotone/convex functions and their links withcompletely positive maps. Nevertheless the proofs very often rely on quite cleverbut simple arguments. As an illustration of the available tools, a very classicalresult is Kadison’s inequality [13] saying that if Φ : A → M is a unital positive(linear) map between C ∗ -algebras, then for a self-adjoint element A in A ,Φ( A ) Φ( A ) . Taking Φ : M n ⊕ M n → M n , Φ( A, B ) = ( A + B ) /
2, this reflects the operatorconvexity of t t . One can think of it as a kind of Jensen’s or Cauchy-Schwarz’sinequality. The main motivation of this paper is to try to get comparison relationsbetween the images of the powers of A . At a first glance, one does not expect tohave many positive results beyond operator convexity. But surprisingly, we noticehere that | Φ( A p )Φ( A q ) | Φ( A p + q ) provided that 0 p q and A >
0. This andsome variations are our concern of the first section.The second section deals more generally with monotone pairs, in place of pairs( A p , A q ). These are pairs ( A, B ) of positive operators, characterized by joint re-lations A = f ( C ) and B = g ( C ) for some C > M n and two non-decreasing,non-negative functions f ( t ) and g ( t ) on [0 , ∞ ). Comparing Φ( A )Φ( B ) with Φ( AB )is non-commutative versions of the classical Chebyshev’s inequality, (cid:16) n n X i =1 a i (cid:17) · (cid:16) n n X i =1 b i (cid:17) n n X i =1 a i b i for non-negative increasing sequences { a i } and { b i } (here Φ is just a state). Date : February 28, 2010.2000
Mathematics Subject Classification.
Primary 15A60, 47A30, 47A60 .
Key words and phrases.
Operator inequalities, Matrix geometric mean, Positive linear maps.Research of both authors was supported by ANR 06-BLAN-0015.
In the last part, we point out the links between complex interpolation and powermeans. Along with some very classical approach and an idea of [11], this is used tofurnish a simple proof of Furuta’s inequality, which is the main tool in section 1.We assume that the reader is familiar with basic notions in operator and matricialinequalities theories. When possible, we state the results in their general context,that is, for von Neumann or C ∗ -algebras. But matrix inequalities for positive linearmaps are essentially finite dimensional results, especially when it comes to unitarycongruences. So, the reader may like to think of the algebras as M n .1. Kadison’s asymmetric type inequalities
In this section, we deal with positive linear maps Φ :
A → M between twounital C ∗ -algebra A and M with units denoted by I . In fact, we may assume that A is the unital C ∗ -algebra generated by a single positive operator A ; hence, by aclassical dilation theorem of Naimark (see [14], Theorem 3.10), our maps Φ will beautomatically completely positive. We will also always assume that these maps areunital, Φ( I ) = I , or more generally sub-unital, Φ( I ) I .Kadison’s inequality is one of the most basic and useful results for such sub-unitalmaps; it states that for any A ∈ A sa (the self-adjoint elements in A ),Φ( A ) Φ( A ) . More generally, if f is operator convex on an interval containing 0 and f (0) f (Φ( A )) Φ( f ( A ))for all A ∈ A sa with spectrum in the domain of f . If we drop the conditionthat 0 is in the domain of f and f (0)
0, this Jensen’s inequality remains truefor unital maps. When Φ is the compression map to a subspace, it is then abasic characterization of operator convexity due to Davis [8]. The general case wasnoted in an influential paper of Choi [6]; nowadays everything is very clear usingStinespring’s theorem (see [14], Theorem 4.1) for completely positive maps.First examples of operator convex/concave functions on R + are given by powers,we refer to the corresponding Jensen inequalities as Choi’s inequality; for A ∈ A + (the positive cone of A ), Φ( A p ) Φ( A ) p , p , and Φ( A ) p Φ( A p ) , p . In the spirit of operator convexity, one can naturally think of looking for morecomparison relations between powers of A .We start with an asymmetric extension of Kadison’s inequality : Theorem 1.1.
Let A ∈ A + and p q . Then, | Φ( A p )Φ( A q ) | Φ( A p + q ) . Proof.
We will derive this result from Furuta’s inequality that we recall as follows :
Let X > Y > in some B ( H ) , let α > and β > . Then, for γ > ( α +2 β ) / (1+2 β ) , X ( α +2 β ) /γ > ( X β Y α X β ) /γ , with equality if and only if X = Y . N ASYMMETRIC KADISON’S INEQUALITY 3
We will discuss about it in Section 3. Now, set X = Φ( A q ) pq , Y = Φ( A p ) . By Choi’s inequality, X > Y . Then we apply Furuta’s inequality to X and Y with α = 2 , β = qp , γ = 2 . (Note that γ = 2 > q/p )1 + 2( q/p ) = α + 2 β β so that assumptions of Furuta’s inequality are satisfied.) Thus we obtain { Φ( A q ) pq } q/p )2 > (cid:16) { Φ( A q ) pq } qp { Φ( A p ) } { Φ( A q ) pq } qp (cid:17) / , equivalently(1.1) Φ( A q ) p/q > | Φ( A p )Φ( A q ) | . Since 1 p/q
2, using once again Choi’s inequality for operator convexfunctions,(1.2) Φ( A p + q ) > Φ( A q ) p/q . Combining (1.1) and (1.2) completes the proof. (cid:3)
Remark . Actually, we have shown the stronger inequality (1.1) that can be re-stated as follows : For 0 α
1, Φ( A ) α > | Φ( A α )Φ( A ) | . Remark.
For 0 < p q , the equality case in Theorem 1.1 entails the equality casein Choi’s inequality, so that Φ( A t ) = Φ( A ) t for all t >
0, in other words A is in themultiplicative domain of Φ. Corollary 1.2.
Assume that moreover M is a von Neumann algebra, then for A ∈ A + and p, q > , there is a partial isometry V ∈ M such that | Φ( A p )Φ( A q ) | V Φ( A p + q ) V ∗ . If M is finite, then V can be chosen to be unitary. This follows from Theorem 1.1 and the polar decomposition. Indeed, for any Z ∈ M , there is a partial isometry V so that Z = V | Z | and moreover | Z ∗ | = V | Z | V ∗ and | Z | = V ∗ | Z ∗ | V . If M is finite, then V can also be chosen unitary. But ingeneral, one can not assume V to be unitary. Remark.
Fix 0 < q < p . Let A = ε ε εε ε εε ε ε , B = and let Φ be the Schur product with B . Then for ε small enough, it follows fromtedious computations on derivatives, that we can not get rid of V in Corollary 1.2like in Theorem 1.1.From now on, we come back to the setting of matrix inequalities where M = M n for some positive integer n . JEAN-CHRISTOPHE BOURIN AND ´ERIC RICARD
The next two results are variations of Corollary 1.2. We rely on an easy conse-quence of the min-max principle; if A > B > M n and f ( t ) is non-decreasingon [0 , ∞ [, then f ( A ) > V f ( B ) V ∗ for some unitary V ∈ M n . Proposition 1.3.
Let A ∈ A + and p, q, r > such that min { p, r } q/ and max { p, r } q . Then, for some unitary V ∈ M n , | Φ( A p )Φ( A q )Φ( A r ) | V Φ( A p + q + r ) V ∗ . Proof.
We may assume q = 1 and r /
2. We then haveΦ( A r ) Φ( A r ) Φ( A ) r so that Φ( A r ) = Φ( A ) r K for a contraction K . Hence, for some unitary U , | Φ( A p )Φ( A )Φ( A r ) | U | Φ( A p )Φ( A ) r | U ∗ so(1.3) | Φ( A p )Φ( A )Φ( A r ) | U (cid:0) { Φ( A ) } r { Φ( A p ) } { Φ( A ) } r (cid:1) U ∗ . Now, observe that a byproduct of Furuta’s inequality is: If X > Y > and, α, β > , then for some unitary W , X α +2 β > W ( X β Y α X β ) W ∗ . Applying this inequality to X = Φ( A ) p and Y = Φ( A p ) with α = 2 , β = (1 + r ) /p and combining with (1.3) yields | Φ( A p )Φ( A )Φ( A r ) | V (Φ( A ) p + r ) V ∗ for some unitary V . Since, by a byproduct of Choi’s inequality, we also have someunitary V such that Φ( A ) p + r V Φ( A p + r ) V ∗ , we get the conclusion. (cid:3) At the cost of one more unitary congruence, assumptions of Proposition 1.3can be relaxed. We will use an inequality of Bhatia and Kittaneh (see [4] for anelementary proof) : For all
A, B in some finite von Neumann algebra M , there issome unitary U ∈ M such that | AB ∗ | U | A | + | B | U ∗ . Proposition 1.4.
Let A > in A and let p, q, r > with q > p, r . Then, for someunitaries U, V in M n , | Φ( A p )Φ( A q )Φ( A r ) | U Φ( A p + q + r ) U ∗ + V Φ( A p + q + r ) V ∗ . Proof.
We may assume q = 1. Let α ∈ [0 ,
1] and note that by Bhatia-Kittaneh’sinequality, | Φ( A p )Φ( A )Φ( A r ) | = | Φ( A p )Φ( A ) α · Φ( A ) − α Φ( A r ) | W | Φ( A p )Φ( A ) α | + | Φ( A r )Φ( A ) − α | W ∗ (1.4)for some unitary W . Then set α = ( r − p + 1) / α A p ) Φ( A ) p and N ASYMMETRIC KADISON’S INEQUALITY 5 Φ( A r ) Φ( A ) r . For the first summand, there are some unitaries W and W suchthat | Φ( A p )Φ( A ) α | = { Φ( A ) p } r − p +12 p { Φ( A p ) } { Φ( A ) p } r − p +12 p W Φ( A ) p + r W ∗ W Φ( A p + r ) W ∗ (1.5)where the last step follows from Choi’s inequality. We also have a unitary W suchthat(1.6) | Φ( A r )Φ( A ) − α | W Φ( A p + r ) W ∗ and combining (1.4), (1.5) and (1.6) completes the proof. (cid:3) Matrix monotony inequalities
Here we try to understand the results of the first section, using the more generalnotion of a monotone pair. Recall that (
A, B ) is said to be a monotone pair in M n if there exist a positive element C ∈ M n and two non-negative, non-decreasingfunctions f and g so that A = f ( C ) and B = g ( C ). A typical example is ( A p , A q )for A > p, q > M n , as many arguments rely on themin-max principle. For instance, we use the following result of [5] which comparesthe singular values of AEB and
ABE for some projections E . Theorem 2.1.
Let ( A, B ) be a monotone pair and let E be a self-adjoint projection.Then, for some unitary V , | AEB | V | ABE | V ∗ . As consequences, we have the following Chebyshev’s type eigenvalue inequalitiesfor compressions [5], λ j [( EAE )( EBE )] λ j [ EABE ]and(2.1) λ j [( EAE )( EBE )( EAE )] λ j [ EABAE ]where λ j [ · ] stands for the list of eigenvalues arranged in decreasing order with theirmultiplicities. Let Φ : M n → M d be a unital completely positive (linear) map. Itis well known (Stinespring) that Φ can be decomposed as Φ( A ) = Eπ ( A ) E , where π : M n → M m is a ∗ -representation (with m n d ) and E ∈ M m is a rank d projection (and identifying E M m E with M d ). Taking into account that we startfrom a commutative C ∗ -algebra, (2.1) is then equivalent to : Corollary 2.2.
Let ( A, B ) be a monotone pair in M n and let Φ : M n → M d be aunital positive map. Then, for some unitary V ∈ M d , Φ( A )Φ( B )Φ( A ) V Φ( ABA ) V ∗ . In the case of pairs of positive powers ( A p , A q ), such results are easy consequencesof Furuta’s inequality. To apply Corollary 2.2 we define a special class of monotonepairs (of positive operators). Definition.
A monotone pair (
A, B ) is concave if A = h ( B ) for some concavefunction h : [0 , ∞ ) → [0 , ∞ ). JEAN-CHRISTOPHE BOURIN AND ´ERIC RICARD
This class contains pairs of powers ( A p , A q ) with 0 p q and we note thatCorollaries 2.4-2.5 below are variations of Theorem 1.1. We first state a factorizationresult. Theorem 2.3.
Let ( A, B ) be a concave monotone pair in M n and let Φ : M n → M d be a unital positive map. Then, for some contraction K and unitary U in M d , Φ( B )Φ( A ) = p Φ( AB ) K p Φ( AB ) U. Proof.
By a continuity argument we may assume that A is invertible, hence (cid:18) AB BB BA − (cid:19) > . Replacing Φ by Φ ◦ E , where E is the conditional expectation onto the C ∗ -algebragenerated by A and B , we can assume that Φ is completely positive so that we get (cid:18) Φ( AB ) Φ( B )Φ( B ) Φ( BA − ) (cid:19) > , equivalently, (cid:18) Φ( AB ) Φ( B )Φ( A )Φ( A )Φ( B ) Φ( A )Φ( BA − )Φ( A ) (cid:19) > . The concavity assumption on (
A, B ) implies that (
A, BA − ) is a monotone pair,indeed both h ( t ) and t/h ( t ) are non-decreasing. By Corollary 2.2, we then have aunitary U such that(2.2) (cid:18) Φ( AB ) Φ( B )Φ( A )Φ( A )Φ( B ) U ∗ Φ( AB ) U (cid:19) > , equivalently, Φ( B )Φ( A ) = p Φ( AB ) LU ∗ p Φ( AB ) U for some contraction L . (cid:3) Theorem 2.3 is equivalent to positivity of the block-matrix (2.2). Consideringthe polar decomposition Φ( A )Φ( B ) = W | Φ( A )Φ( B ) | we infer (cid:0) I − W ∗ (cid:1) (cid:18) Φ( AB ) Φ( B )Φ( A )Φ( A )Φ( B ) U ∗ Φ( AB ) U (cid:19) (cid:18) I − W (cid:19) > Corollary 2.4.
Let ( A, B ) be a concave monotone pair in M n and let Φ : M n → M d be a unital positive map. Then, for some unitary V ∈ M d , | Φ( A )Φ( B ) | Φ( AB ) + V Φ( AB ) V ∗ . Recall that a norm is said symmetric whenever k U AV k = k A k for all A and allunitaries U, V . Corollary 2.4 yields for concave monotone pairs some Chebyshev’stype inequalities for symmetric norms, k Φ( A )Φ( B ) k k Φ( AB ) k . N ASYMMETRIC KADISON’S INEQUALITY 7
It is not clear that this can be extended to all monotone pairs. In fact, for concavemonotone pairs, Theorem 2.3 entails a stronger statement. Given
X, Y >
0, recallthat the weak log-majorization relation X ≺ wlog Y means Y j k λ j [ X ] Y j k λ j [ Y ]for all k = 1 , , · · · . This entails k X k k Y k for all symmetric norms. Theorem 2.3and Horn’s inequality yield : Corollary 2.5.
Let ( A, B ) be a concave monotone pair in M n and let Φ : M n → M d be a unital, positive linear map. Then, | Φ( A )Φ( B ) | ≺ wlog Φ( AB ) . In case of pairs ( A p , A q ) we have more : Proposition 2.6.
Let A > in M n , let p, q > and let Φ as above. Then, for alleigenvalues, λ j [Φ( A p )] λ j [Φ( A q )] λ j [Φ( A p + q )] . Proof.
We outline an elementary proof. It suffices to show that for a given projec-tion E ,(2.3) λ j [ EA p E ] λ j [ EA q E ] λ j [ EA p + q E ] . In case of the first eigenvalue, this can be written via the operator norm k · k ∞ as(2.4) k EA p E k ∞ k EA q E k ∞ k EA p + q E k ∞ . The proof of (2.4) follows from Young’s and Jensen’s inequalities (always true forthe operator norm), k EA p E k ∞ k EA q E k ∞ pp + q k EA p E k p + qp ∞ + qp + q k EA q E k p + qq ∞ k EA p + q E k ∞ . The min-max characterization of eigenvalues combined with (2.4) implies the propo-sition; indeed simply take Q a projection commuting with E of corank j − k QEA p + q EQ k ∞ = λ j [ EA p + q E ] and apply (2.4) with QE instead of E . (cid:3) Results of this section follow from (2.1), equivalently from Corollary 2.2, andhence have been stated for unital positive maps. In fact these results can be statedto all sub-unital positive maps. In particular the key Corollary 2.2 becomes :
Corollary 2.2a.
Let ( A, B ) be a monotone pair in M n and let Φ : M n → M d be asub-unital positive map. Then, for some unitary V ∈ M d , Φ( A )Φ( B )Φ( A ) V Φ( ABA ) V ∗ . Proof.
Let A be the unital ∗ -algebra generated by A and B . Restricting Φ to A ,it follows from Stinespring’s theorem (or from Naimark’s theorem) that Φ can bedecomposed as Φ( A ) = Zπ ( A ) Z , where π : A → M m is a ∗ -representation (with m nd ) and Z ∈ M m is a positive contraction (and identifying E M m E with M d for some projection E > Z ). JEAN-CHRISTOPHE BOURIN AND ´ERIC RICARD
Since ( π ( A ) , π ( B )) is monotone, it then suffices to prove the result for congruencemaps of M n of type Φ( X ) = ZXZ where Z is a positive contraction. We may thenderive the result from (2.1) and a two-by-two trick : Note that A = (cid:18) A
00 0 (cid:19) and B = (cid:18) B
00 0 (cid:19) form a monotone pair. Note also that E = (cid:18) Z ( Z ( I − Z )) / ( Z ( I − Z )) / I − Z (cid:19) is a projection. By (2.1), λ j [( EA E )( EB E )( EA E )] λ j [ EA B A E ] , equivalently,(2.5) λ j [ | EA EB / | ] λ j [( A B A ) / E ( A B A ) / ] . Observe that(2.6) | EA EB / | = (cid:18) B / ZAZAZB /
00 0 (cid:19) ≃ (cid:18) Z / AZBZAZ /
00 0 (cid:19) where ≃ means unitary equivalence, and similarly,(2.7) ( A B A ) / E ( A B A ) / ≃ (cid:18) Z / ABAZ /
00 0 (cid:19) . Combining (2.6) and (2.7) with (2.5) and replacing Z / by Z yields ZAZ · ZBZ · ZAZ V ( ZABAZ ) V ∗ for some unitary V . (cid:3) We end Section 2 with a remark about the two-by-two trick used to deriveCorollary 2.4. This can be used to get some triangle type matrix inequalities. Forinstance, given two operators A and B in some von Neumann algebra M , thereexists a partial isometry V such that:(2.8) | A + B | | A | + | B | + V ∗ ( | A ∗ | + | B ∗ | ) V . To check it, note that, since for all X , (cid:18) | X ∗ | XX ∗ | X | (cid:19) > , we thus have for all V , (cid:0) − V ∗ I (cid:1) (cid:18) | A ∗ | + | B ∗ | A + BA ∗ + B ∗ | A | + | B | (cid:19) (cid:18) − VI (cid:19) > , and taking V the partial isometry in the polar decomposition of A + B yields (2.8).This can be used to give a very short proof of the triangle inequality for the tracenorm in semi-finite von Neumann algebras. N ASYMMETRIC KADISON’S INEQUALITY 9 Means and order preserving relations
Furuta’s inequality was used as key tool in the first section; here we present apossible proof for completeness. For that purpose we use the geometric mean of pos-itive definite matrices and Ando-Hiai’s inequality. We do not pretend to originalityand we closely follow an approach due to Ando, Hiai, Fujii and Kamei. However,we point out an interesting observation connecting the geometric mean to complexinterpolation. In fact this observation is rather old : Identifying positive operatorswith quadratic forms, it is worth noting that Donoghue’s construction with complexinterpolation [9] seems to be the first appearance of the matrix geometric mean.In the whole section we consider B ( H ), the set of all bounded operators on aHilbert space H , and its positive invertible part, B + .For details and some important results around the geometric mean we refer to[1], [2], references herein, and [4] for a nice survey of other features of the weightedgeometric means, especially as geodesics on the cone of positive operators.3.1. Means and interpolation.
Let α ∈ [0 ,
1] and consider a map B + × B + → B + ( A, B ) A♯ α B satisfying the two natural requirements for an α -geometrical mean1. If AB = BA then A♯ α B = A − α B α .2. ( X ∗ AX ) ♯ α ( X ∗ BX ) = X ∗ ( A♯ α B ) X for any invertible X .Choosing the appropriate X , we necessarily have A♯ α B = A / ( A − / BA − / ) α A / . So there is a unique extension of the α -geometrical mean for commuting operatorswhich is invariant under congruence, that is called the α -geometrical mean.Matrix geometric means have their roots in the work of Pusz and Woronowicz[15] about functional calculus for sesquilinear forms. Their construction is closelyrelated to complex interpolation. Coming back to means, these links are evenclearer.We briefly recall the complex interpolation method of Calderon, see [3] for acomplete exposition.Two Banach spaces A and A are said to be an interpolation couple if there isanother Banach space V and continuous embeddings A i → V . So we have a way toidentify elements and it makes sense to speak of A ∩ A and A + A (which arealso Banach spaces with the usual norms). The idea of interpolation is to assignfor each α ∈ [0 ,
1] a space that is intermediate between the A i . The construction isa bit technical.Let ∆ = { z ∈ C | < Re z < } , δ i = { z ∈ C | Re z = i } for i = 0, 1. Define F ( A , A ) as the set of maps f : ∆ → A + A , such thati) f is analytic in ∆.ii) for i = 0, 1, f ( δ i ) ⊂ A i and f : δ i → A i is bounded and continuous.iii) for i = 0, 1, lim t ∈ R →±∞ || f ( i + i t ) || A i = 0.Equipped with the norm || f || = max i =0 , { sup z ∈ δ i || f ( z ) || A i } F ( A , A ) becomes a Banach space. Finally for α ∈ [0 , A , A ) α = { x ∈ A + A : ∃ f ∈ F ( A , A ) so that f ( α ) = x } with the quotient norm || x || ( A ,A ) α = inf {|| f || : f ( α ) = x } . This functor has many nice properties. The most common is the interpolationprinciple; consider two interpolation couples ( A , A ) and ( B , B ) and boundedmaps T i : A i → B i so that T and T coincide on A ∩ A , then one can define amap T α : ( A , A ) α → ( B , B ) α which extends T i on A ∩ A , and moreover onehas || T α || || T || − α || T || α .There are concrete examples where these interpolated norms can be computed.If A = L ∞ ([0 , A = L ([0 , A , A ) α = L /α ([0 , A ∈ B + , then it defines an equivalent hilbertian norm on H by || h || A = || A / h || H . And conversely any equivalent hilbertian norm on H arises from some A ∈ B + . We denote by H A the Hilbert space coming from A .Now take A i ∈ B + , ( H A , H A ) forms an interpolation couple of Hilbert space(with the obvious identification). The resulting interpolated space for α ∈ [0 , H , associated to an operator that we call A α .Let’s have a look at the properties of ( A , A ) I α ( A , A ) = A α .First, it is an easy exercise to check that if A and A commute then I α ( A , A ) = A α = A − α A α .Secondly, let X ∈ B ( H ) be invertible. With B i = X ∗ A i X , it is clear that X : H B i → H A i is a unitary for i = 0, 1. From the interpolation principle, X willalso be unitary for the interpolated norms. Coming back to operators, this saysthat I α ( X ∗ A X, X ∗ A X ) = X ∗ I α ( A , A ) X .So we can conclude that the α -geometric mean is the interpolation functor ofindex α . With this in mind, all properties of the means come from basic results inthe complex interpolation theory.Take ( A , A ) and ( B , B ) in ( B + ) and assume that B i A i . This means thatthe identity of H is a contraction from H A i to H B i . By the interpolation principle,the same holds for the interpolated norms. So we can conclude that the α -meanis monotone. Note that this gives another proof of the monotony of A A α for0 α A i and B i in B + , and 0 < λ < H λA i +(1 − λ ) B i →H λA i ⊕ H (1 − λ ) B i , h ( h, h ) is an isometry. From properties of the interpo-lation functor, we deduce that the same map H ( λA +(1 − λ ) B ) ♯ α ( λA +(1 − λ ) B ) →H λ ( A ♯ α A ) ⊕ H (1 − λ )( B ♯ α B ) is a contraction. Coming back to an inequality onoperators gives the concavity. This illustrates the well-known fact that taking sub-spaces and interpolation do not commute.Another useful result is the reiteration theorem : For any α, β, γ ∈ [0 , A ∩ A is dense in both A and A )(( A , A ) α , ( A , A ) β ) γ = ( A , A ) (1 − γ ) α + γβ . N ASYMMETRIC KADISON’S INEQUALITY 11
This means that for any x, y, z ∈ [0 ,
1] and
A, B ∈ B + ,( A♯ x B ) ♯ z ( A♯ y B ) = A♯ x (1 − z )+ yz B. Of course this can also be checked directly from the formulae defining ♯ .The next theorem is the Ando-Hiai inequality. We only use the language ofoperator mean, but this is really a proof in the spirit of the interpolation theory. Theorem 3.1.
Let
A, B ∈ B + and < s < . Then, k ( A♯ α B ) s k ∞ k A s ♯ α B s k ∞ . Proof.
By homogeneity we may assume k A♯ α B k ∞ = 1. Hence we have A♯ α B I .By using monotony of geometric means and the reiteration principle we then get A s ♯ α B s = ( I♯ s A ) ♯ α ( I♯ s B ) > (( A♯ α B ) ♯ s A ) ♯ α (( A♯ α B ) ♯ s B )= (( A♯ α B ) ♯ s ( A♯ B )) ♯ α (( A♯ α B ) ♯ s ( A♯ B ))= ( A♯ α (1 − s ) B ) ♯ α ( A♯ α (1 − s )+ s B ) = A♯ α B. Thus k A s ♯ α B s k ∞ > (cid:3) Remark . A theory of complex interpolation for families of Banach spaces has beendeveloped in [7]. The family may be indexed by the unit circle in C , say A ( z ), withsome technical assumptions. The interpolation then provides a family of spaces A ( z ) for | z | <
1. This can be used define a mean of several operators. For instance,in the case of n operators, one may pick a partition of the unit circle in n sets E i with Lebesgue measure α i , and choose the family A ( z ) = H A i if z ∈ E i . Then theinterpolated space at 0 is of the form A (0) = H A , and one may think of A as a( α i )-mean of the A i ’s. Unfortunately this definition depends on the choice of the E i (unless n = 2). This kind of approach for interpolation of a finite family ofspaces can also be found in [10].3.2. From means to order relations.
Next we explain how to go from Ando-Hiai’s inequality to Furuta’s theorem (their equivalence was pointed out in [11]).Let
A, B ∈ B + with A > B . Then, A − ♯ / B I , so by Ando-Hiai’s inequality, A − p ♯ / B p I whenever p >
1. Equivalently we have an order preserving relationfor f ( t ) = t p with p > A p > ( A p/ B p A p/ ) / , p > . Such inequalities suggest to look for the best exponents p, r, w for which(3.1) A > B > ⇒ A ( p + r ) w > ( A r/ B p A r/ ) w and consequently to get interesting substitutes to the lack of operator monotony of f ( t ) = t p , p > A > B > ⇒ A − r ♯ α B p I. Because of homogeneity, this can hold only for α = rp + r . If p
1, this inequality isobvious by the monotony of the mean. For p >
1, as above thanks to Ando-Hiai’s inequality, one only need to find s A − sr ♯ α B sp I ; we’ve just said that s = 1 /p works. We have proved : Lemma 3.2.
Let
A, B ∈ B + with A > B and p, r > . Then, A − r ♯ rp + r B p I. Taking another mean with B p , we obtain the optimal form of (3.2) : Lemma 3.3.
Let
A, B ∈ B + with A > B and r > , p > . Then, A − r ♯ rp + r B p B A. Hence we have recaptured quite easily two lemmas due to Fujii and Kamei [11].We come back to relations of the form (3.1), the last lemma says :(3.3) A > B > ⇒ A r > ( A r B p A r ) rp + r , r > , p > . Equivalently, A ( p + r ) w > ( A r B p A r ) w , r > , p > , where w = rp + r . This is still valid for w rp + r by the operator monotony of t t α ,0 α
1. We obtain Furuta’s theorem :
Theorem 3.4.
Let
A, B > in B ( H ) and r > , p > . If q > ( p + r ) / (1 + r ) ,then A ( p + r ) /q > ( A r B p A r ) /q . The general statement follows from the case B ∈ B + by continuity.3.2.1. Comments.
Around 1985 it was conjectured by Kwong that A > B > A > ( AB A ) / , equivalently A > | BA | . In 1987, Furuta [12] proved hisinequality. Some numerical experiments lead him to know the condition on theexponents and he obtained a direct, quite ingenious proof. However the naturalconjecture of Kwong may be written via geometric means and is nicely answeredby a basic case of Ando-Hiai’s inequality (1994). Hence order preserving relationsmay be obtained from a study of weighted geometric means. We have followed thisidea, mainly developed by Ando, Hiai, Fujii and Kamei. References
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Functional calculus for sesquilinear forms and the purificationmap , Rep. Mathematical Phys. (1975), no. 2, 159-170. Laboratoire de math´ematiques, Universit´e de Franche-Comt´e, 16 route de Gray,25030 Besanc¸on, France
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