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DEMONSTRATIO MATHEMATICA
Vol. 49
No 4
2016
Ismail Nikoufar
A PERSPECTIVE APPROACH FOR CHARACTERIZATION
OF LIEB CONCAVITY THEOREM
Communicated by M. Bożejko
Abstract. Lieb’s extension theorem holds for generalized p ` q P r0, 1s and Ando
convexity theorem holds for q ´ r ą 1. In this paper, we give a complete characterization
for concavity or convexity of Lieb well known theorem in the case where p ` q ≥ 1 or
p ` q ≤ 0. We also characterize some auxiliary results including Ando theorem for q ´ r ≤ 1.
1. Introduction
Matrix convexity and more generally noncommutative functional analysis
provides an appropriate framework for many of the calculations in quantum
information theory. This theory also provides nonclassical techniques that
clarify some of the conceptual problems in matrix convexity theory [E].
Throughout this paper, we assume that f and h are real–valued continuous
functions on the closed interval I. We say that f is matrix (operator) convex
if
f pcA1 ` p1 ´ cqA2 q ≤ cf pA1 q ` p1 ´ cqf pA2 q
for all self–adjoint n ˆ n matrices A1 , A2 with spectra in I, and all c P r0, 1s.
Also, f is matrix concave if ´f is matrix convex. For a self–adjoint matrix
A, the value f pAq is defined by the spectral theorem.
We remarked in [ENE] a beautiful contribution without any commutativity
assumption and defined a fully noncommutative generalized perspective of
two variables (associated to f and h) as follows:
Pf ∆h pA, Bq :“ hpBq1{2 f phpBq´1{2 AhpBq´1{2 qhpBq1{2 ,
where A, B are self–adjoint and strictly positive matrices, respectively with
2010 Mathematics Subject Classification: Primary 81P45; Secondary 15A39, 15A42,
47A63, 81R15.
Key words and phrases: perspective function, generalized perspective function, matrix
convex function.
DOI: 10.1515/dema-2016-0040
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I. Nikoufar
spectra in the closed interval I containing 0. In a natural way, when h is
the identity map, we achieve the matrix version of a fully noncommutative
perspective of two variables (associated to f ) as follows:
Pf pA, Bq :“ B 1{2 f pB ´1{2 AB ´1{2 qB 1{2 .
For some applications of the notion of perspective, we refer the readers to
[N] and references therein.
Let P be a perspective function. We say that P is jointly convex if
P pcA1 ` p1 ´ cqA2 , cB1 ` p1 ´ cqB2 q ≤ cP pA1 , B1 q ` p1 ´ cqP pA2 , B2 q
for all self–adjoint matrices A1 , A2 , strictly positive matrices B1 , B2 , and
c P r0, 1s. Also, P is jointly concave if ´P is jointly convex.
We proved in [ENE] the necessary and sufficient conditions for joint
convexity of a fully noncommutative generalized perspective function. For
the next applications, we now refine [ENE, Theorem 2.5]:
Theorem 1. Suppose that f and h are continuous functions and h ą 0.
(i) If f is matrix convex pconcaveq with f p0q ≤ 0 (f p0q ≥ 0) and h is matrix
concave, then the generalized perspective function Pf ∆h is jointly convex
pconcaveq.
(ii) If the generalized perspective function Pf ∆h is jointly convex pconcaveq,
then f is matrix convex pconcaveq.
(iii) If f p0q ą 0 and the generalized perspective function Pf ∆h is jointly
convex pconcaveq, then h is matrix convex pconcaveq.
(iv) If f p0q ă 0 and the generalized perspective function Pf ∆h is jointly
convex pconcaveq, then h is matrix concave pconvex q.
In particular, by using the affine version of Hansen–Pedersen–Jensen
inequality [H, Theorem 2.1] we could prove the following theorem (see [ENE,
Theorem 2.2, Corollary 2.3]):
Theorem 2. The function f is matrix convex pconcaveq if and only if the
perspective function Pf is jointly convex pconcaveq.
Lieb’s concavity theorem holds for generalized p ` q P r0, 1s. We investigated [NEE] a different and simple proof for Lieb concavity and Ando
convexity theorem based on the pα, βq-geometric mean. In this paper, we
extend a fact enounced in [NEE] concerning the notion of pα, βq-geometric
mean and prove its reverse. We apply this fact to complete the characterization of Lieb well known theorem in the case where p ` q ≥ 1 or p ` q ≤ 0. We
also characterize some auxiliary results including Ando convexity theorem
for q ´ r ≤ 1. Indeed, our aim is to find conditions implying the convexity of
Lieb’s function and the concavity of Ando’s function.
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A perspective approach for characterization of Lieb theorem
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2. Characterization of Lieb and Ando well known theorems
We first recall the following remark from [B, ch. V].
Remark 1. On the positive half line p0, `8q the function f ptq “ tr , where
r is a real number, is matrix concave if and only if r P r0, 1s. If r ≥ 0, then f
is matrix convex if and only if r P r1, 2s. The function f is matrix convex if
r P r´1, 0s and it is not matrix convex for r P p´8, ´1q Y p0, 1q Y p2, `8q.
Kubo and Ando [KA] defined α-geometric mean of two strictly positive
matrices for 0 ≤ α ≤ 1 as follows:
1
1
1
1
A7α B :“ A 2 pA´ 2 BA´ 2 qα A 2 .
It is obvious that the λ-geometric mean is perspective of the function tλ in
the sense that
(1)
A7λ B “ Ptλ pB, Aq.
For λ P r´1, 0s Y r1, 2s, the matrix convexity of tλ implies the joint convexity
of the λ-geometric mean A7λ B. Also, in the case where λ P r0, 1s, the matrix
concavity of tλ gives the joint concavity of the λ-geometric mean. However, by
using Theorem 2, Remark 1 and equality (1), we can conclude the following
theorem.
Theorem 3. Assume that λ is a real number.
(i) If λ ≥ 0, then the λ-geometric mean is jointly convex if and only if
λ P r1, 2s.
(ii) The λ-geometric mean is jointly concave if and only if λ P r0, 1s.
(iii) The λ-geometric mean is jointly convex if λ P r´1, 0s.
(iv) The λ-geometric mean is not jointly convex if λ P p´8, ´1q Y p0, 1q Y
p2, `8q.
In his celebrated paper [L] Lieb proved that
(2)
pA, Bq ÞÑ T raceAq K ˚ B p K
is jointly concave on the strictly positive n ˆ n matrices for any n ˆ n matrix
K when p, q ą 0 and p ` q ≤ 1. It solves the convexity question of the
Wigner–Yanase–Dyson skew information [WY].
A significant complement of Lieb’s concavity theorem is Ando’s convexity
theorem [A1] stating that the function
(3)
pA, Bq ÞÑ T raceAq K ˚ B ´r K
is jointly convex on the strictly positive n ˆ n matrices, where 1 ≤ q ≤ 2 and
0 ≤ r ≤ 1 with q ´ r ą 1.
We are worth noting that an elegant and simple way of obtaining joint
concavity of (2) and joint convexity of (3) was given by Nikoufar et al. [NEE]
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as follows:
(4)
xRB #pλ1 ,γ1 q LA pK ˚ q, K ˚ y “ T raceAq K ˚ B p K,
(5)
xRB #pλ2 ,γ2 q LA pK ˚ q, K ˚ y “ T raceAq K ˚ B ´r K
for some λ1 , γ1 , γ2 P r0, 1s and λ2 P r1, 2s, where xA, By “ T raceAB ˚ is
Hilbert–Schmidt inner product and LA , RB are left and right multiplication
operators. One may wish to find a similar applications and it would be
interesting to find applications of these new family of generalized perspective
functions.
Some natural questions now arise: What can we say about joint convexity
or concavity of Lieb extension theorem in the case where p ` q ≥ 1 or
p ` q ≤ 0? What can we say about joint convexity or concavity of Ando
theorem in the case where q ´ r ≤ 1? We now proceed to discuss on these
questions.
We first need to extend a fact from [NEE]. Recently, Nikoufar et al. [NEE]
introduced the notion of operator pα, βq-geometric mean as a generalization
of the notion of operator α-geometric mean on strictly positive matrices for
0 ≤ α ≤ 2 and 0 ≤ β ≤ 1 as follows:
β
β
β
β
A#pα,βq B :“ A 2 pA´ 2 BA´ 2 qα A 2 .
Note that, in this sense, every operator α-geometric mean is an operator
pα, 1q-geometric mean. Although, the theory of α-geometric mean is wellestablished and we can write A#pα,βq B “ Aβ #α B and although, it is easy
to deduce some properties of pα, βq-geometric mean from that of α-mean, we
will discuss on α, β, where we give a complete characterization of some well
known theorems including Lieb and Ando theorems. The following theorem
is a more generalized version of [NEE, Theorem 1.1].
Theorem 4. Assume that λ, γ are real numbers.
(i) The operator pλ, γq-geometric mean is jointly concave if λ, γ P r0, 1s.
(ii) The operator pλ, γq-geometric mean is jointly convex if λ P r1, 2s and
γ P r0, 1s.
Proof. The proof follows from Theorem 1 (i) and the fact that
(6)
A#pλ,γq B “ Ptλ ∆tγ pB, Aq.
Remark 2. For the function f ptq “ tλ , the condition f p0q ≤ 0 does not
hold in the case where λ P r´1, 0q and so Theorem 1 (i) can not be applied.
In this case, however, joint convexity of pλ, γq-geometric mean is doubtful.
We may consider a reverse version of above theorem as follows.
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A perspective approach for characterization of Lieb theorem
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Theorem 5. Assume that λ, γ are real numbers.
(i) If the operator pλ, γq-geometric mean is jointly convex, then λ P r1, 2s.
(ii) If the operator pλ, γq-geometric mean is jointly concave, then λ P r0, 1s.
Proof. The proof follows from Theorem 1 (ii), Remark 1, and equality (6).
Leib’s Theorem states that if 0 ă s ă 1, then the function
F pA, Bq “ T raceAs K ˚ B 1´s K
(7)
is jointly concave on the strictly positive matrices A, B (see a different and
simple proof in [NEE]). The joint convexity or joint concavity of this function
can be characterized with respect to all real numbers s.
Theorem 6. Let s be a real number.
(i)
(ii)
(iii)
(iv)
The function (7) is jointly convex if s P r´1, 0s Y r1, 2s.
The function (7) is jointly concave if s P r0, 1s.
The function (7) is not jointly convex if s P p´8, ´1q Y p0, 1q Y p2, `8q.
The function (7) is not jointly concave if s P p´8, 0q Y p1, `8q.
Proof. We have
xRB #s LA pK ˚ q, K ˚ y “ T raceAs K ˚ B 1´s K.
Note that part (i) follows from Theorem 3 (i). Parts (ii) and (iii) follow from
Theorem 3 (ii). Parts (iii) and (iv) are a simple consequence of (i) and (ii).
On the other hand, Leib’s extension theorem states that for generalized
p ` q ≤ 1, the function (2) is jointly concave on the strictly positive matrices
(see the elegant proofs in [A2], [E] and a simple proof in [NEE]). In the
following theorems, we prove that Lieb’s function (2) is jointly convex if
p ` q ≥ 1.
Theorem 7. Assume that p, q are real numbers.
(i) The function (2) is jointly concave if q P r0, 1q and q ≤ p ` q ≤ 1.
(ii) The function (2) is jointly convex if q P p1, 2s and 1 ≤ p ` q ≤ q.
Proof. By a simple calculation we see that
xRB #pq,
p
q
1´q
LA pK ˚ q, K ˚ y “ T raceAq K ˚ B p K.
Apply now Theorem 4 (i) to conclude part (i) and Theorem 4 (ii) to deduce
part (ii).
Theorem 8. Assume that p, q are real numbers.
(i) The function (2) is jointly concave if p P r0, 1q and p ≤ p ` q ≤ 1.
(ii) The function (2) is jointly convex if p P p1, 2s and 1 ≤ p ` q ≤ p.
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Proof. A simple verification shows
xLA #pp,
q
q
1´p
RB pK ˚ q, K ˚ y “ T raceAq K ˚ B p K.
By using Theorem 4 (i) we get part (i) and by applying Theorem 4 (ii) we
obtain part (ii).
We now characterize convexity or concavity of Ando function (3). We
prove that Ando’s function (3) is jointly concave if q ´ r ≤ 1 and jointly
convex if q ´ r ≥ 1.
Theorem 9. Assume that q, r are real numbers.
(i) The function (3) is jointly concave if q P r0, 1q and q ≤ q ´ r ≤ 1.
(ii) The function (3) is jointly convex if q P p1, 2s and 1 ≤ q ´ r ≤ q.
Proof. We can see, by an easy computation, that
˚
˚
q ˚ ´r
r
xRB #pq, q´1
q LA pK q, K y “ T raceA K B K.
The results follow from Theorem 7, by setting p :“ ´r.
Theorem 10. Assume that q, r are real numbers.
(i) The function (3) is jointly concave if r P p´1, 0s and ´r ≤ q ´ r ≤ 1.
(ii) The function (3) is jointly convex if r P r´2, ´1q and 1 ≤ q ´ r ≤ ´r.
Proof. A simple computation entails
xLA #p´r,
q
q
1`r
RB pK ˚ q, K ˚ y “ T raceAq K ˚ B ´r K.
The desired results follow from Theorem 8, by replacing p :“ ´r.
We know that the function
pA, Bq ÞÑ T raceAq K ˚ B 1´r K
(8)
is jointly convex on the set of strictly positive matrices when 1 ă r ≤ q ≤ 2
(see [NEE]). In the following corollaries we complete this result.
Corollary 1. Assume that q, r are real numbers.
(i) The function (8) is jointly concave if q P r0, 1q and q ≤ r ≤ 1.
(ii) The function (8) is jointly convex if q P p1, 2s and 1 ≤ r ≤ q.
Proof. We know that
xRB #pq, 1´r q LA pK ˚ q, K ˚ y “ T raceAq K ˚ B 1´r K.
1´q
Apply now Theorem 4 (i) to see part (i) and Theorem 4 (ii) to obtain
part (ii).
Corollary 2. Assume that q, r are real numbers.
(i) The function (8) is jointly concave if r P p0, 1s and 0 ≤ q ≤ r.
(ii) The function (8) is jointly convex if r P r´1, 0q and r ≤ q ≤ 0.
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A perspective approach for characterization of Lieb theorem
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Proof. We have
xLA #p1´r, q q RB pK ˚ q, K ˚ y “ T raceAq K ˚ B 1´r K.
r
Consequently, part (i) follows from Theorem 4 (i) and Theorem 4 (ii) implies
part (ii).
Acknowledgement. I would like to express my sincere thanks to Professor Elliott H. Lieb for his valuable remarks after having read my first
draft.
References
[HD]
[A1]
R. Hill, A. Dow, An index formula, J. Differential Equations 15 (1982), 197–211.
T. Ando, Concavity of certain maps on positive definite matrices and applications
to Hadamard products, Linear Algebra Appl. 26 (1979), 203–241.
[A2] J. S. Aujla, A simple proof of Lieb concavity theorem, J. Math. Phys. 52, 043505
(2011), 3 pages. doi:10.1063/1.3573594
[B]
R. Bhatia, Matrix Analysis, Graduate Texts in Mathematics 169, Springer, New
York, 1997.
[ENE] A. Ebadian, I. Nikoufar, M. Eshagi Gordji, Perspectives of matrix convex functions,
Proc. Nat. Acad. Sci. U.S.A. 108(18) (2011), 7313–7314.
E. G. Effros, A matrix convexity approach to some celebrated quantum inequalities,
[E]
Proc. Nat. Acad. Sci. U.S.A. 106(4) (2009), 1006–1008.
[H]
F. Hansen, G. Pedersen, Jensen’s operator inequality, Bull. London Math. Soc.
35 (2003), 553–564.
[KA] F. Kubo, T. Ando, Means of positive linear operators, Math. Ann. 246 (1979–1980),
205–224.
[L]
E. Lieb, Convex trace functions and the Wigner–Yanase–Dyson conjecture, Adv.
Math. 11 (1973), 267–288.
[N]
I. Nikoufar, On operator inequalities of some relative operator entropies, Adv. Math.
259 (2014), 376–383.
[NEE] I. Nikoufar, A. Ebadian, M. Eshagi Gordji, The simplest proof of Lieb concavity
theorem, Adv. Math. 248 (2013), 531–533.
[WY] E. P. Wigner, M. M. Yanase, Information contents of distributions, Proc. Nat.
Acad. Sci. U.S.A. 49 (1963), 910–918.
I. Nikoufar
DEPARTMENT OF MATHEMATICS
PAYAME NOOR UNIVERSITY
P.O. BOX 19395-3697 TEHRAN, IRAN
E-mail: [email protected]
Received March 9, 2015
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