Table Of ContentNON-COMMUTATIVE f-DIVERGENCE FUNCTIONAL
MOHAMMAD SAL MOSLEHIAN AND MOHSEN KIAN
Abstract. We introduce the non-commutative f-divergence functional Θ(A,B):=
13 T Bt12f Bt−12AtBt−21 Bt12dµ(t)foranoperatorconvexfunctionf,whereA=e(Aet)t∈T
0 Rand B =(cid:16)(Bt)t∈T are(cid:17)continuous fields of Hilbert space operatorsand study its prop-
2 e
erties. We establish some relations between the perspective of an operator convex
n e
function f and the non-commutative f-divergence functional. In particular, an op-
a
J erator extension of Csisza´r’s result regarding f-divergence functional is presented.
0
As some applications, we establish a refinement of the Choi–Davis–Jensen operator
3
inequality, obtain some unitarily invariant norm inequalities and give some results
]
related to the Kullback–Leibler distance.
A
F
.
h
t
a
m 1. Introduction and Preliminaries
[
Let B(H ) be the algebra of all bounded linear operators on a complex Hilbert space
1
v H and I denote the identity operator. If dimH = n, we identify B(H ) with the
9
algebra M (C) of all n×n matrices with entries in the complex number field C. An
4 n
3
operator A is said to be positive (denoted by A ≥ 0) if hAx,xi ≥ 0 for all vectors
7
. x ∈ H . If, in addition, A is invertible, then it is called strictly positive (denoted by
1
0
A > 0). By A ≥ B we mean that A−B is positive, while A > B means that A−B
3
1 is strictly positive. A map Φ on B(H ) is called positive if Φ(A) ≥ 0 for each A ≥ 0.
:
v An operator C is called an isometry if C∗C = I, a contraction if C∗C ≤ I and an
i
X expansive operator if C∗C ≥ I.
r
a A continuous real valued function f defined on an interval J is said to be operator
convex if
f(λA+(1−λ)B) ≤ λf(A)+(1−λ)f(B),
for all self-adjoint operators A,B with spectra contained in J and any λ ∈ [0,1]. If −f
is operator convex, then f is said to be operator concave. Let J and J be two real
1 2
intervals. A jointly operator convex function is a function f defined on J × J such
1 2
2010 Mathematics Subject Classification. 47A63, 46L05,26D15, 15A60, 60E15.
Key words and phrases. Information theory; Kullback–Leibler distance; f-divergence functional;
Csisza´r’s result; perspective function; operator convex.
1
2 M.S. MOSLEHIAN,M. KIAN
that
f(λ(A,B)+(1−λ)(C,D)) ≤ λf(A,B)+(1−λ)f(C,D),
for allself-adjoint operatorsA,C with spectra contained inJ , all self-adjoint operators
1
B,D with spectra contained in J and all λ ∈ [0,1]; see e.g. [16] for the definition of
2
f(A,B).
The Jensen operator inequality, due to Hansen and Pedersen states that f : J → R
is operator convex if and only if
f(C∗AC) ≤ C∗f(A)C, (1.1)
for any isometry C and any self-adjoint operator A with spectrum contained in J,
see [10] for various equivalent assertions. If 0 ∈ J and f(0) ≤ 0, then f is operator
convex on J if and only if (1.1) holds for any contraction C. Some other various
characterizations of operator convexity can be found in [10, Chapter 1]; see also [18, 19]
and references therein.
The Choi–Davis–Jensen inequality states that if f is operator convex, then
f(Φ(A)) ≤ Φ(f(A)), (1.2)
for any unital positive linear map Φ and any self-adjoint operator A, whose spectrum
is contained in the domain of f; see [20] for a characterization for the case of equality.
An extension of this significant inequality reads as follows.
Theorem A.[17]Let f be an operator convex function on J and Φ ,··· ,Φ be positive
1 n
linear maps on B(H ) with n Φ (I) = I. Then
i=1 i
Pn n
f Φ (A ) ≤ Φ (f(A )), (1.3)
i i i i
!
i=1 i=1
X X
for all self-adjoint operators A (i = 1,··· ,n) with spectra contained in J. In partic-
i
ular,
n n
f C∗A C ≤ C∗f(A )C , (1.4)
i i i i i i
!
i=1 i=1
X X
whenever n C∗C = I.
i=1 i i
Let T be a locally compact Hausdorff space and A be a C∗-algebra of Hilbert space
P
operators. A field (A ) of operators in A is said to be continuous if the function
t t∈T
t 7→ A is norm continuous on T. Moreover, If µ is a Radon measure on T and the
t
function t 7→ A is integrable on T, then the Bochner integral A dµ(t) is defined to
t T t
R
NON-COMMUTATIVE f-DIVERGENCE FUNCTIONAL 3
be the unique element of A with the property that
ρ A dµ(t) = ρ(A )dµ(t),
t t
(cid:18)ZT (cid:19) ZT
for any linear functional ρ in the norm dual A∗ of A.
Furthermore, let A and B be C∗-algebras of operators. A field (Φ ) : A → B of
t t∈T
positive linear maps is said to be continuous if the function t 7→ Φ (A) is continuous on
t
T for every A ∈ A. If the C∗-algebras A and B are unital and the function t 7→ Φ (I)
t
is integrable on T with integral I, then we say that the field (Φ ) is unital.
t t∈T
The following result, is the Jensen operator inequality for continuous fields of oper-
ators.
Theorem B. [11] Let f be an operator convex function defined on an interval J, and
let A and B be unital C∗-algebras. If (Φ ) : A → B is a unital field of positive linear
t t∈T
maps defined on a locally compact Hausdorff space T with a bounded Radon measure
µ, then
f Φ (A )dµ(t) ≤ Φ (f(A ))dµ(t), (1.5)
t t t t
(cid:18)ZT (cid:19) ZT
for every norm bounded continuous field (A ) of self-adjoint operators in A with
t t∈T
spectra contained in J.
Let f be a convex function on a convex set K ⊆ R. Following [12], the perspective
function g associated to f is defined on the set {(x,y) : y > 0 and x ∈ K} by
y
x
g(x,y) := yf .
y
(cid:18) (cid:19)
As an operator extension of the perspective function, Effros [9] introduced the per-
spective function of an operator convex function f by
L
g(L,R) := Rf ,
R
(cid:18) (cid:19)
for commuting strictly positive operators L and R and proved the following notable
theorem.
Theorem C. [9] If f is operator convex, when restricted to the commuting strictly
positive operators, then the perspective function (L,R) 7→ g(L,R) = Rf L is jointly
R
operator convex.
(cid:0) (cid:1)
He also extended the generalized perspective function, defined by Mar´echal [14, 15]
to operators. Given continuous functions f and h and commuting strictly positive
operators L and R, Effros defined the operator extension of the generalized perspective
4 M.S. MOSLEHIAN,M. KIAN
function by
L
(f∆h)(L,R) := h(R)f ,
h(R)
(cid:18) (cid:19)
and proved the following assertion.
Theorem D. If f is operator convex with f(0) ≤ 0 and h is operator concave with
h > 0, then f∆h is jointly convex on commuting strictly positive operators.
The authors of [8] extended Effros results by removing the restriction to commuting
operators and proved analogue results to Theorem C and Theorem D.
An interesting study of such functions for operators was introduced by Kubo and
Ando. They considered the case where f is an operator monotone function and es-
tablished a relation between the operator monotone functions and the operator means
(see [10, Chapter 5]).
One of the most principal matters in applications of probability theory is to find a
suitable measure between two probability distributions. Many kinds of such measures
have been studied and applied in several fields such as signal processing, genetics and
economics. One of the most famous of such measures is the Csisza´r f-divergence
functional, which includes several measures.
For a convex function f : [0,∞) → R, Csisza´r [4, 5] introduced the f-divergence
functional by
n
p
i
I (p,q) := q f , (1.6)
f i
q
i=1 (cid:18) i(cid:19)
X
for positive n-tuples p = (p ,···e,pe ) and q = (q ,··· ,q ), in which undefined expres-
1 n 1 n
sions were interpreted by
e e
0 p p f(t)
f(0) = lim f(t), 0f = 0, 0f = lim f = p lim .
t→0+ 0 0 ǫ→0+ ǫ t→∞ t
(cid:18) (cid:19)
(cid:16) (cid:17) (cid:16) (cid:17)
Also Csisza´r and Ko¨rner [6] obtained the following result.
Theorem E. If f : [0,∞) → R is convex, then I (p,q) is jointly convex in p and q
f
and
n n p e e e e
q f i=1 i ≤ I (p,q) (1.7)
i n q f
i=1 (cid:18)Pi=1 i(cid:19)
X
for all positive n-tuples p = (p ,··· ,Pp ),q = (q ,···e,qe).
1 n 1 n
A series of results and inequalities related to f-divergence functionals can be found
in [1, 3, 7, 13]. e e
The paper is organized as follows. In section 2, we introduce an operator extension
of f-divergence functional for an operator convex function f, which is more general
NON-COMMUTATIVE f-DIVERGENCE FUNCTIONAL 5
than the perspective function associated to f. We give some properties of our non-
commutative f-divergence functional andestablish itsrelationshiptotheperspective of
f. In particular, an operator extension of (1.7) is presented. In section 3, we provide
some applications for our results. More precisely, a refinement of the Choi–Davis–
Jensen operator inequality is obtained, some unitarily invariant norm inequalities are
presented and some results related to the Kullback–Leibler distance are given.
2. Non-commutative f-divergence functionals
Throughout this section, assume that T is a locally compact Hausdorff space with a
bounded Radon measure µ and A and B are C∗-algebras of Hilbert space operators.
Assume that A = (A ) and B = (B ) are continuous fields of self-adjoint and
t t∈T t t∈T
strictly positive operators in A, respectively, and (Φ ) : A → B is a unital field of
t t∈T
e e
positive linear maps. Furthermore, when T is the finite set {1,··· ,n} and µ is the
counting measure, suppose that L = (L ,··· ,L ) and R = (R ,··· ,R ) are n-tuples
1 n 1 n
of self-adjoint and strictly positive operators on a Hilbert space H , respectively, and
(Φ ,··· ,Φ ) is an n-tuple of posietive linear maps on B(eH ).
1 n
Let f : [0,∞) → R be a convex function. The perspective function g associated to
f is defined by
g(L,R) := R12f(R−21LR−12)R21,
where L is a self-adjoint operator and R is a strictly positive operator on a Hilbert
space H . In [8] it is shown that f is operator convex if and only if g is jointly
operator convex. We consider a more general case. Let us define the non-commutative
f-divergence functional Θ by
1 −1 −1 1
Θ(A,B) := B2f B 2A B 2 B2dµ(t). (2.1)
t t t t t
ZT (cid:16) (cid:17)
Hence, in the discrete ceasee Θ is defined by
n
1 −1 −1 1
Θ(L,R) = R2f(R 2L R 2)R2. (2.2)
i i i i i
i=1
X
e e
By the same argument as in [8], it is easy to see that Θ is jointly operator convex if
and only if f is operator convex. In the sequel, we study some properties of Θ and
establish some relations between Θ and g. The following result is an extension of (1.7).
Theorem 2.1. Let f be an operator convex function, and g be the corresponding per-
spective function. Then
g(A,B) ≤ Θ(A,B), (2.3)
e e
6 M.S. MOSLEHIAN,M. KIAN
where A = A dµ(t) and B = B dµ(t).
T t T t
R R
Proof.
f B−21AB−12
(cid:16) (cid:17) −1 −1
2 2
= f B dµ(s) A dµ(t) B dµ(s)
s t s
(cid:18)ZT (cid:19) ZT (cid:18)ZT (cid:19) !
−1 −1
2 2
= f B dµ(s) A B dµ(s) dµ(t)
s t s
ZT (cid:18)ZT (cid:19) (cid:18)ZT (cid:19) !
−1 −1
2 1 −1 −1 1 2
= f B dµ(s) B2 B 2A B 2 B2 B dµ(s) dµ(t)
s t t t t t s
ZT (cid:18)ZT (cid:19) (cid:16) (cid:17) (cid:18)ZT (cid:19) !
−1 −1
2 1 −1 −1 1 2
≤ B dµ(s) B2f B 2A B 2 B2 B dµ(s) dµ(t)
s t t t t t s
ZT (cid:18)ZT (cid:19) (cid:16) (cid:17) (cid:18)ZT (cid:19)
( by the Jensen operator inequality (1.5))
−1 −1
2 1 −1 −1 1 2
= B dµ(s) B2f B 2A B 2 B2dµ(t) B dµ(s)
s t t t t t s
(cid:18)ZT (cid:19) ZT (cid:16) (cid:17) (cid:18)ZT (cid:19)
= B−21Θ(A,B)B−12,
(cid:3)
whence we obtain the desired inequality (2.3).
e e
Corollary 2.2. Let f be an operator convex function and g be the perspective function
of f. Then
(i) The perspective function g of an operator convex function f is sub-additive.
More general,
n n n
g L , R ≤ g(L ,R ). (2.4)
i i i i
!
i=1 i=1 i=1
X X X
(ii) f ( n L ) ≤ n g(L ,R ), whenever n R = I.
i=1 i i=1 i i i=1 i
P P P
Let T and T be disjoint locally compact Hausdorff spaces and T = T ∪T . The
1 2 1 2
following refinement of (2.3) holds.
Corollary 2.3. Let g be the perspective function of an operator convex function f.
Then
1
2g (A,B) ≤ g(A ,B )+g(A ,B ) ≤ Θ(A,B), (2.5)
2 T1 T1 T2 T2
(cid:18) (cid:19)
where A = A dµ(t), B = B dµ(t), A = A dµ(t) andeB e= B dµ(t).
T t T t T1 T1 t T1 T1 t
R R R R
NON-COMMUTATIVE f-DIVERGENCE FUNCTIONAL 7
Proof. Since (A,B) = (A ,B )+(A ,B ), the first inequality of (2.5) follows from
T1 T1 T2 T2
the joint convexity of g. The second inequality follows immediately from Theorem
(cid:3)
2.1.
Theorem 2.4. Let L and R (1 ≤ i,j ≤ n) be self-adjoint and strictly positive
ij ij
operators, respectively, and let p (1 ≤ j ≤ n) be positive numbers. If f is operator
j
convex, then
n n
g(L ,R ) ≤ p Θ(Li,Ri),
i i i
i=1 i=1
X X
e e
where L = n p L , R = n p R , Li = (L ,··· ,L ), Ri = (R ,··· ,R ).
i j=1 j ij i j=1 j ij i1 in i1 in
P P
Proof. Using (2.3) for A and B (1 ≤ i ≤en) we obtain e
i i
g(L ,R ) = R21f R−21L R−21 R21 ≤ Θ(pLi,pRi), (1 ≤ i ≤ n), (2.6)
i i i i i i i
(cid:16) (cid:17)
where pAi = (p A ,··· ,p A ) and pBi = (p B ,e··· ,ep B ). In addition,
1 i1 n in 1 i1 n in
n
eΘ(pLi,pRi) = (pjRij)12fe (pjRij)−21(pjLij)(pjRij)−21 (pjRij)21
Xj=1 (cid:16) (cid:17)
e e n
1 −1 −1 1
= p R2f(R 2L R 2)R2. (2.7)
j ij ij ij ij ij
j=1
X
Summing (2.6) over i we get
n n
g(L ,R ) ≤ Θ(pLi,pRi)
i i
i=1 i=1
X X
n n e e
1 −1 −1 1
= p R2f(R 2L R 2)R2 (by (2.7))
j ij ij ij ij ij
i=1 j=1
XX
n n
1 −1 −1 1
= p R2f(R 2L R 2)R2
j ij ij ij ij ij
j=1 i=1
X X
n
= p Θ(Li,Ri).
j
j=1
X
e e (cid:3)
Forcontinuousfunctionsf andhandcommuting matricesLandR, Effros[9]defined
the function (L,R) 7→ (f∆h)(L,R) by
L
(f∆h)(L,R) := f h(R).
h(R)
(cid:18) (cid:19)
8 M.S. MOSLEHIAN,M. KIAN
He also proved that if f is operator convex with f(0) ≤ 0 and h is operator concave
with h > 0, then f∆h is jointly operator convex. In [8], definition and properties of
f∆h were naturally given for two not necessarily commuting self-adjoint operators.
Let f and h be continuous real valued functions defined on an interval J and µ be
a probability measure on T. As a generalization of f∆h, we define f∇h by
(f∇h)(A,B) := h(Bt)21f h(Bt)−21Ath(Bt)−12 h(Bt)12dµ(t).
ZT (cid:16) (cid:17)
e e
It is not hard to see that f is operator convex with f(0) ≤ 0 and h is operator concave
with h > 0 if and only if f∇h is jointly operator convex.
The next result, is a Choi–Davis–Jensen type inequality for f∆h.
Theorem 2.5. Let f be an operator convex function with f(0) ≤ 0 and h be an operator
concave function with h > 0. If Φ (I)dµ(t) ≤ I, then
T t
R
(f∆h) Φ (A )dµ(t), Φ (B )dµ(t) ≤ Φ ((f∆h)(A ,B ))dµ(t). (2.8)
t t t t t t t
(cid:18)ZT ZT (cid:19) ZT
In particular, if g is the perspective function of f, then
g Φ (A )dµ(t), Φ (B )dµ(t) ≤ Φ (g(A ,B ))dµ(t), (2.9)
t t t t t t t
(cid:18)ZT ZT (cid:19) ZT
where B is strictly positive for any t ∈ T.
t
Proof. Let (B ) be a continuous filed of self-adjoint operators. Define the field of
t t∈T
positive linear maps Ψ : A → B by
t
−1 −1
2 1 1 2
Ψt(X) = h Φt(Bt)dµ(t) Φt h(Bt)2Xh(Bt)2 h Φt(Bt)dµ(t) .
(cid:18)ZT (cid:19) (cid:16) (cid:17) (cid:18)ZT (cid:19)
Since h is operator concave, h > 0 and Φ (I)dµ(t) ≤ I, we have
T t
R
Φ (h(B ))dµ(t) ≤ h Φ (B )dµ(t) .
t t t t
ZT (cid:18)ZT (cid:19)
Therefore
−1 −1
2 2
Ψ (I)dµ(t) = h Φ (B )dµ(t) Φ (h(B ))h Φ (B )dµ(t) ≤ I.
t t t t t t t
ZT ZT (cid:18)ZT (cid:19) (cid:18)ZT (cid:19)
NON-COMMUTATIVE f-DIVERGENCE FUNCTIONAL 9
Put C = Φ (B )dµ(t). We have
T t t
R
(f∆h) Φ (A )dµ(t), Φ (B )dµ(t)
t t t t
(cid:18)ZT ZT (cid:19)
= h(C)21f h(C)−12 Φt(At)dµ(t)h(C)−21 h(C)21
(cid:18) ZT (cid:19)
= h(C)12f h(C)−21 Φt(At)h(C)−12dµ(t) h(C)21
(cid:18)ZT (cid:19)
= h(C)12f Ψt h(Bt)−21 Ath(Bt)−12 dµ(t) h(C)21
(cid:18)ZT (cid:16) (cid:17) (cid:19)
≤ h(C)12 Ψt f h(Bt)−21 Ath(Bt)−21 dµ(t)h(C)21
ZT (cid:16) (cid:16) (cid:17)(cid:17)
(by the Jensen operator inequality (1.5))
= Φ ((f∆h)(A ,B ))dµ(t),
t t t
ZT
(cid:3)
which gives rise to (2.8).
As special cases of Theorem 2.5 we have the following result.
Corollary 2.6. Let f be an operator convex function with f(0) ≤ 0 and h be an
operator concave function with h > 0. If Φ is a positive linear map on B(H ) with
Φ(I) ≤ I, then
(f∆h)(Φ(A),Φ(B)) ≤ Φ((f∆h)(A,B)), (2.10)
for all self-adjoint operators A,B. In particular, if g is the perspective function asso-
ciated to f, then
g(Φ(A),Φ(B)) ≤ Φ(g(A,B)), (2.11)
for any self-adjoint operator A and any strictly positive operator B.
Example 2.7. Let Φ be a positive linear map on B(H ). Applying Corollary 2.6 to
the operator convex function f(t) = tβ (−1 ≤ β ≤ 0 or 1 ≤ β ≤ 2) and the operator
concave function h(t) = tα (0 ≤ α ≤ 1), we obtain
Φ(B)α2 Φ(B)−α2Φ(A)Φ(B)−2α βΦ(B)α2 ≤ Φ Bα2 B−α2AB−α2 βBα2 . (2.12)
(cid:16) (cid:17) (cid:16) (cid:17)
In particular, for α = 1 and β = −1, inequality (2.12) g(cid:0)ives rise to (cid:1)
2
Φ(B)12Φ(A)−1Φ(B)12 ≤ Φ B21A−1B12 .
(cid:16) (cid:17)
Note that with α = 1 and β = −1, inequality (2.12) gives rise to the known inequality
Φ(B)Φ(A)−1Φ(B) ≤ Φ BA−1B .
(cid:0) (cid:1)
10 M.S. MOSLEHIAN,M. KIAN
Corollary 2.8. Let f be an operator convex function with f(0) ≤ 0 and h be an
operator concave function with h > 0. Then
(f∆h)(hAx,xi,hBx,xi) ≤ h(f∆h)(A,B)x,xi,
for all self-adjoint operators A,B ∈ B(H ) and all unit vectors x ∈ H . In particular,
if g is the perspective function of f, then
g(hAx,xi,hBx,xi) ≤ hg(A,B)x,xi, (2.13)
for any self-adjoint operator A ∈ B(H ), any strictly positive operator B ∈ B(H ) and
any unit vector x ∈ H .
Theorem 2.9. Let f and f be operator convex functions with f (0) ≤ 0 and f (0) ≤ 0
1 2 1 2
and let h be an operator concave function with h > 0. The following assertions are
equivalent:
(1) f ≤ f ;
1 2
(2) (f ∆h) Φ (A )dµ(t), Φ (B )dµ(t)) ≤ Φ ((f ∆h)(A ,B ))dµ(t) for all
1 T t t T t t T t 2 t t
unital fields (Φ ) and all continuous fields of operators (A ) and (B ) ;
(cid:0)R t t∈T R (cid:1) R t t∈T t t∈T
(3) f Φ (A )dµ(t) ≤ Φ (f (A ))dµ(t) for all continuous fields of operators
1 T t t T t 2 t
(A ) .
(cid:0)tRt∈T (cid:1) R
Proof. (1) ⇒ (2) Let f ≤ f and (B ) be a continuous field of self-adjoint operators
1 2 t t∈T
in A. Let C = Φ (B )dµ(t). Define the field of positive linear maps (Ψ ) : A → B
T t t t t∈T
by
R
Ψt(X) = h(C)−21Φt h(Bt)12Xh(Bt)21 h(C)−21.
(cid:16) (cid:17)
It follows from the operator concavity of h and h > 0 that
Φ (h(B ))dµ(t) ≤ h Φ (B )dµ(t) .
t t t t
ZT (cid:18)ZT (cid:19)
Therefore
Ψt(I)dµ(t) = h(C)−21Φt(h(Bt))h(C)−21 ≤ I.
ZT ZT
