vol22_pp179-190. 139KB Jun 04 2011 12:06:12 AM
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 22, pp. 179-190, March 2011
ELA
http://math.technion.ac.il/iic/ela
A DIAZ–METCALF TYPE INEQUALITY FOR POSITIVE LINEAR
MAPS AND ITS APPLICATIONS∗
MOHAMMAD SAL MOSLEHIAN† , RITSUO NAKAMOTO‡ , AND YUKI SEO§
Abstract. We present a Diaz–Metcalf type operator inequality as a reverse Cauchy–Schwarz
inequality and then apply it to get some operator versions of P´
olya–Szeg¨
o’s, Greub–Rheinboldt’s,
Kantorovich’s, Shisha–Mond’s, Schweitzer’s, Cassels’ and Klamkin–McLenaghan’s inequalities via
a unified approach. We also give some operator Gr¨
uss type inequalities and an operator Ozeki–
Izumino–Mori–Seo type inequality. Several applications are included as well.
Key words. Diaz–Metcalf type inequality, Reverse Cauchy–Schwarz inequality, Positive map,
Ozeki–Izumino–Mori–Seo inequality, Operator inequality.
AMS subject classifications. 46L08, 26D15, 46L05, 47A30, 47A63.
1. Introduction. The Cauchy–Schwarz inequality plays an essential role in
mathematical analysis and its applications. In a semi-inner product space (H , h·, ·i)
the Cauchy–Schwarz inequality reads as follows
|hx, yi| ≤ hx, xi1/2 hy, yi1/2
(x, y ∈ H ).
There are interesting generalizations of the Cauchy–Schwarz inequality in various
frameworks, e.g., finite sums, integrals, isotone functionals, inner product spaces, C ∗ algebras and Hilbert C ∗ -modules; see [5, 6, 7, 9, 11, 13, 17, 20] and references therein.
There are several reverses of the Cauchy–Schwarz inequality in the literature: Diaz–
Metcalf’s, P´olya–Szeg¨
o’s, Greub–Rheinboldt’s, Kantorovich’s, Shisha–Mond’s, Ozeki–
Izumino–Mori–Seo’s, Schweitzer’s, Cassels’ and Klamkin–McLenaghan’s inequalities.
Inspired by the work of J.B. Diaz and F.T. Metcalf [4], we present several reverse
Cauchy–Schwarz type inequalities for positive linear maps. We give a unified treatment of some reverse inequalities of the classical Cauchy–Schwarz type for positive
∗ Received by the editors on September 29, 2010. Accepted for publication on February 19, 2011
Handling Editor: Bit-Shun Tam.
† Department of Pure Mathematics, Centre of Excellence in Analysis on Algebraic Structures
(CEAAS), Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran (moslehian@
ferdowsi.um.ac.ir, [email protected]). Supported by a grant from Ferdowsi University of Mashhad
(No. MP89144MOS).
‡ Faculty of Engineering, Ibaraki University, Hitachi, Ibaraki 316-0033, Japan (r-naka@net1.
jway.ne.jp).
§ Faculty of Engineering, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitamacity, Saitama 337-8570, Japan ([email protected]).
179
Electronic Journal of Linear Algebra ISSN 1081-3810
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180
M.S. Moslehian, R. Nakamoto, and Y. Seo
linear maps.
Throughout the paper B(H ) stands for the algebra of all bounded linear operators
acting on a Hilbert space H . We simply denote by α the scalar multiple αI of the
identity operator I ∈ B(H ). For self-adjoint operators A, B the partially ordered
relation B ≤ A means that hBξ, ξi ≤ hAξ, ξi for all ξ ∈ H . In particular, if 0 ≤ A,
then A is called positive. If A is a positive invertible operator, then we write 0 < A. A
linear map Φ : A → B between C ∗ -algebras is said to be positive if Φ(A) is positive
whenever A is. We say that Φ is unital if Φ preserves the identity. The reader is
referred to [9, 19] for undefined notations and terminologies.
2. Operator Diaz–Metcalf type inequality. We start this section with our
main result. Recall that the geometric operator mean A ♯ B for positive operators
A, B ∈ B(H ) is defined by
1
21 1
1
1
A ♯ B = A 2 A− 2 BA− 2
A2
if 0 < A.
Theorem 2.1. Let A, B ∈ B(H ) be positive invertible operators and Φ :
B(H ) → B(K ) be a positive linear map.
(i) If m2 A ≤ B ≤ M 2 A for some positive real numbers m < M , then the
following inequalities hold:
• Operator Diaz–Metcalf inequality of first type
M mΦ(A) + Φ(B) ≤ (M + m)Φ(A♯B) ;
• Operator Cassels inequality
M +m
Φ(A)♯Φ(B) ≤ √
Φ(A♯B) ;
2 Mm
• Operator Klamkin–McLenaghan inequality
Φ(A♯B)
−1
2
Φ(B)Φ(A♯B)
−1
2
√
√
1
1
− Φ(A♯B) 2 Φ(A)−1 Φ(A♯B) 2 ≤ ( M − m)2 ;
• Operator Kantorovich inequality
Φ(A)♯Φ(A−1 ) ≤
M 2 + m2
.
2M m
(ii) If m21 ≤ A ≤ M12 and m22 ≤ B ≤ M22 for some positive real numbers
m1 < M1 and m2 < M2 , then the following inequalities hold:
Electronic Journal of Linear Algebra ISSN 1081-3810
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A Diaz–Metcalf Type Inequality for Positive Linear Maps
• Operator Diaz–Metcalf inequality of second type
m2
M2
M 2 m2
Φ(A) + Φ(B) ≤
+
Φ(A♯B) ;
M 1 m1
m1
M1
• Operator P´
olya–Szeg¨
o inequality
!
r
r
M1 M2
m1 m2
1
Φ(A♯B) ;
+
Φ(A)♯Φ(B) ≤
2
m1 m 2
M 1 M2
• Operator Shisha–Mond inequality
Φ(A♯B)
−1
2
Φ(B)Φ(A♯B)
−1
2
1
1
− Φ(A♯B) 2 Φ(A)−1 Φ(A♯B) 2
!2
r
r
M2
m2
≤
−
;
m1
M1
• Operator Gr¨
uss type inequality
2
√
√
√
M1 M2
M 1 M 2 − m 1 m2
M1 M2
.
min
,
Φ(A)♯Φ(B) − Φ(A♯B) ≤
√
2 m1 m2
m 1 m2
Proof.
(i) If m2 A ≤ B ≤ M 2 A for some positive real numbers m < M , then
−1
−1
m ≤ A 2 BA 2 ≤ M 2 .
2
(ii) If m21 ≤ A ≤ M12 and m22 ≤ B ≤ M22 for some positive real numbers
m1 < M1 and m2 < M2 , then
m2 =
−1
−1
m22
M22
2 BA 2 ≤
≤
A
= M2 .
M12
m21
(2.1)
In any case we then have
1/2 −1
1/2
−1
−1
−1
A 2 BA 2
M − A 2 BA 2
− m ≥ 0,
whence
Mm + A
Hence
−1
2
BA
−1
2
12
−1
−1
≤ (M + m) A 2 BA 2
.
−1
21
−1
M mA + B ≤ (M + m)A1/2 A 2 BA 2
A1/2 = (M + m)A♯B .
(2.2)
M mΦ(A) + Φ(B) ≤ (M + m)Φ(A♯B) .
(2.3)
Since Φ is a positive linear map, (2.2) yields the operator Diaz–Metcalf inequality of
first type as follows:
Electronic Journal of Linear Algebra ISSN 1081-3810
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M.S. Moslehian, R. Nakamoto, and Y. Seo
In the case when (ii) holds we get the following, which is called the operator Diaz–
Metcalf inequality of second type:
M 2 m2
m2
M2
Φ(A) + Φ(B) ≤
+
Φ(A♯B) .
M 1 m1
m1
M1
Following the strategy of [21], we apply the operator geometric–arithmetic inequality
to M mΦ(A) and Φ(B) to get:
√
1
M m(Φ(A)♯Φ(B)) = (M mΦ(A))♯Φ(B) ≤ (M mΦ(A) + Φ(B)) .
2
(2.4)
It follows from (2.3) and (2.4) that
M +m
Φ(A♯B) ,
Φ(A)♯Φ(B) ≤ √
2 Mm
which is said to be the operator Cassels inequality under the assumption (i); see also
[16]. Under the case (ii) we can represent it as the following inequality being called
the operator P´
olya–Szeg¨
o inequality or the operator Greub–Rheinboldt inequality:
!
r
r
1
M1 M2
m1 m 2
Φ(A)♯Φ(B) ≤
+
Φ(A♯B) .
(2.5)
2
m1 m 2
M 1 M2
It follows from (2.5) that
!
!
r
M1 M2
m 1 m2
− 1 Φ(A♯B)
+
m1 m2
M1 M2
2
√
M 1 M 2 − m 1 m2
√
Φ(A♯B) .
√
2 m 1 m2 M 1 M 2
1
Φ(A)♯Φ(B) − Φ(A♯B) ≤
2
√
=
r
(2.6)
It follows from (2.1) that
so
1/2
−1
−1
m2
M2
A1/2 ≤
A ≤ A1/2 A 2 BA 2
A,
M1
m1
M 2 M2
m21 m2
≤ A♯B ≤ 1
.
M1
m1
Now, (2.6) and (2.7) yield that
Φ(A)♯Φ(B) − Φ(A♯B) ≤
√
2
√
M1 M2 − m1 m2 M12 M2
√
.
√
m1
2 m1 m2 M 1 M 2
An easy symmetric argument then follows that
2
√
√
√
M1 M2
M 1 M 2 − m 1 m2
M 1 M2
Φ(A)♯Φ(B) − Φ(A♯B) ≤
,
min
,
√
2 m1 m2
m 1 m2
(2.7)
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A Diaz–Metcalf Type Inequality for Positive Linear Maps
presenting a Gr¨
uss type inequality.
If A is invertible and Φ is unital and m21 = m2 ≤ A ≤ M 2 = M12 , then by
putting m22 = 1/M 2 ≤ B = A−1 ≤ 1/m2 = M22 in (2.5) we get the following operator
Kantorovich inequality:
Φ(A)♯Φ(A−1 ) ≤
M 2 + m2
.
2M m
It follows from (2.3) that
Φ(A♯B)
−1
2
Φ(B)Φ(A♯B)
−1
2
−1
1
1
− Φ(A♯B) 2 Φ(A)−1 Φ(A♯B) 2
1
−1
1
≤ M + m − M mΦ(A♯B) 2 Φ(A)Φ(A♯B) 2 − Φ(A♯B) 2 Φ(A)−1 Φ(A♯B) 2
√
1/2
√
−1
−1
≤ M + m − 2 Mm −
M m Φ(A♯B) 2 Φ(A)Φ(A♯B) 2
1/2 2
1
1
− Φ(A♯B) 2 Φ(A)−1 Φ(A♯B) 2
√
√
≤ ( M − m)2 ,
(2.8)
that is, an operator Klakmin–Mclenaghan inequality when (i) holds. Under (ii), we
get the following operator Shisha–Szeg¨
o inequality from (2.8):
!2
r
r
−1
−1
1
1
M
m
2
2
−
.
Φ(A♯B) 2 Φ(B)Φ(A♯B) 2 − Φ(A♯B) 2 Φ(A)−1 Φ(A♯B) 2 ≤
m1
M1
3. Applications. If (a1 , . . . , an ) and (b1 , . . . , bn ) are n-tuples of real numbers
with 0 < m1 ≤ ai ≤ M1 (1 ≤ i ≤ n), 0 < m2 ≤ bi ≤ M2 (1 ≤ i ≤ n), we can
consider the positive linear map Φ(T ) = hT x, xi on B(Cn ) = Mn (C) and let A =
diag(a21 , . . . , a2n ), B = diag(b21 , . . . , b2n ) and x = (1, . . . , 1)t in the operator inequalities
above to get the following classical inequalities:
• Diaz–Metcalf inequality [4]
n
X
k=1
b2k
n
n
m2 X
M2
m2 M 2 X 2
ak ≤
+
+
ak bk .
m1 M 1
m1
M1
k=1
k=1
• P´olya–Szeg¨
o inequality [23]
Pn
Pn
2
2
1
k=1 ak
k=1 bk
≤
Pn
2
4
( k=1 ak bk )
r
M1 M2
+
m1 m 2
r
!2
;
!2
;
m1 m2
M1 M2
• Shisha–Mond inequality [24]
Pn
Pn
a2
k=1 ak bk
Pnk=1 k − P
n
2 ≤
a
b
k
k
k=1
k=1 bk
r
M1
−
m2
r
m1
M2
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• A Gr¨
uss type inequality
n
X
a2k
k=1
!1/2
n
X
k=1
b2k
!1/2
−
n
X
ak bk
k=1
2
√
√
√
M1 M2
M 1 M 2 − m 1 m2
M1 M2
min
,
.
≤
√
2 m1 m 2
m1 m 2
Using the same argument with a positive n-tuple (a1 , . . . , an ) of real numbers with
0 < m ≤ ai ≤ M (1 ≤ i ≤ n), x = √1n (1, . . . , 1)t , we get from Kantorovich inequality
that
• Schweitzer inequality [2]
n
1X 2
a
n i=1 i
!
n
1 X −2
a
n i=1 i
!
≤
(M 2 + m2 )2
.
4M 2 m2
If (a1 , . . . , an ) and (b1 , . . . , bn ) are n-tuples of real numbers with 0 < m ≤ ai /bi ≤
M (1 ≤ i ≤ n), we can consider the positive linear map Φ(T ) = hT x, xi on B(Cn ) =
√
√
Mn (C) and let A = diag(a21 , . . . , a2n ), B = diag(b21 , . . . , b2n ) and x = ( w1 , . . . , wn )t
based on the weight w
¯ = (w1 , . . . , wn ), in the operator inequalities above to get the
following classical inequalities:
• Cassels inequality [25]
Pn
Pn
2
wk a2k k=1 wk b2k
(M + m)
;
≤
Pn
2
4mM
( k=1 wk ak bk )
k=1
• Klamkin–McLenaghan inequality [14]
n
X
k=1
wk a2k
n
X
k=1
wk b2k
−
n
X
wk ak bk
k=1
!2
≤
n
n
√
X
√ 2 X
wk ak bk
M− m
wk a2k .
k=1
k=1
Using the same argument, we obtain a weighted form of the P´olya–Szeg¨o inequality as follows:
• Grueb–Rheinboldt inequality [10]
Pn
2
wk a2k k=1 wk b2k
(M1 M2 + m1 m2 )
≤
.
Pn
2
4m1 m2 M1 M2
( k=1 wk ak bk )
Pn
k=1
One can assert the integral versions of discrete results above by considering
L2 (X, µ), where (X, µ) is a probability space, as a Hilbert space via hh1 , h2 i =
R
h h dµ, multiplication operators A, B ∈ B(L2 (X, µ))) defined by A(h) = f 2 h
X 1 2
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A Diaz–Metcalf Type Inequality for Positive Linear Maps
and B(h) = g 2 h for bounded f, g ∈ L2 (X, µ) and a positive linear map Φ by
R
Φ(T ) = X T (1)dµ on B(L2 (X, µ))). For instance, let us state integral versions of the
Cassels and Klamkin–McLenaghan inequalities. These two inequalities are obtained,
first for bounded positive functions f, g ∈ L2 (X, µ) and next for general positive
functions f, g ∈ L2 (X, µ) as the limits of sequences of bounded positive functions.
Corollary 3.1. Let (X, µ) be a probability space and f, g ∈ L2 (X, µ) with
0 ≤ mg ≤ f ≤ M g for some scalars 0 < m < M . Then
Z
f 2 dµ
X
Z
X
g 2 dµ ≤
(M + m)2
4M m
Z
X
2
f gdµ
and
Z
2
f dµ
X
Z
2
X
g dµ −
Z
X
2
Z
Z
√
√ 2
M− m
f gdµ
f 2 dµ .
f gdµ ≤
X
X
Pn
Considering the positive linear functional Φ(R) = i=1 hRξi , ξi i on B(H ), where
ξ1 , . . . , ξn ∈ H , we get the following versions of the Diaz–Metcalf and P´olya–Szeg¨o
inequalities in a Hilbert space.
Corollary 3.2. Let H be a Hilbert space, let ξ1 , . . . , ξn ∈ H and let T, S ∈
B(H ) be positive operators satisfying 0 < m1 ≤ T ≤ M1 and 0 < m2 ≤ S ≤ M2 .
Then
n
n
n
X
M 2 m2 X
m2 X
M2
kT ξi k2 +
kSξi k2 ≤
+
k(T 2 ♯S 2 )1/2 ξi k2
M1 m1 i=1
m
M
1
1
i=1
i=1
and
n
X
i=1
2
kT ξi k
!1/2
n
X
i=1
1
≤
2
2
kSξi k
r
!1/2
M1 M2
+
m 1 m2
r
m1 m2
M1 M2
!
n
X
i=1
k(T 2 ♯S 2 )1/2 ξi k2 .
4. A Gr¨
uss type inequality. In this section we obtain another Gr¨
uss type
∗
∗
inequality, see also [18]. Let A be a C -algebra and let B be a C -subalgebra of
A . Following [1], a positive linear map Φ : A → B is called a left multiplier if
Φ(XY ) = Φ(X)Y for every X ∈ A , Y ∈ B.
The following lemma is interesting on its own right.
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Lemma 4.1. Let Φ be a unital positive linear map on A , A ∈ A and M, m be
complex numbers such that
Re ((M − A)∗ (A − m)) ≥ 0 .
(4.1)
Then
2
Φ(|A|2 ) − |Φ(A)| ≤
1
|M − m|2 .
4
Proof. For any complex number c ∈ C, we have
Φ(|A|2 ) − |Φ(A)|2 = Φ(|A − c|2 ) − |Φ(A − c)|2 .
(4.2)
Since for any T ∈ A the operator equality
2
M + m
1
2
= Re ((M − T )(T − m)∗ )
|M − m| − T −
4
2
holds, the condition (4.1) implies that
2 !
M
+
m
1
Φ A −
≤ |M − m|2 .
2
4
(4.3)
Therefore, it follows from (4.2) and (4.3) that
Φ(|A|2 ) − |Φ(A)|2 ≤ Φ(|A −
≤
M +m 2
| )
2
1
|M − m|2 .
4
Remark 4.2. If (i) Φ is a unital positive linear map and A is a normal operator
or (ii) Φ is a 2-positive linear map and A is an arbitrary operator, then it follows from
[3] that
2
0 ≤ Φ(|A|2 ) − |Φ(A)| .
(4.4)
Condition (4.4) is stronger than positivity and weaker than 2-positivity; see [8]. Another class of positive linear maps satisfying (4.4) are left multipliers, cf. [1, Corollary
2.4].
Lemma 4.3. Let a positive linear map Φ : A → B be a unital left multiplier.
Then
2
|Φ(A∗ B) − Φ(A)∗ Φ(B)| ≤
Φ(|A|2 ) − |Φ(A)|2
Φ(|B|2 ) − |Φ(B)|2
(4.5)
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A Diaz–Metcalf Type Inequality for Positive Linear Maps
Proof. If we put [X, Y ] := Φ(X ∗ Y ) − Φ(X)∗ Φ(Y ), then A is a right pre-inner
product C ∗ -module over B, since Φ(X ∗ Y ) is a right pre-inner product B-module, see
[1, Corollary 2.4]. It follows from the Cauchy–Schwarz inequality in pre-inner product
C ∗ -modules (see [15, Proposition 1.1]) that
2
|Φ(A∗ B) − Φ(A)∗ Φ(B)| = [B, A][A, B]
≤ k[A, A]k[B, B]
= kΦ(A∗ A) − Φ(A)∗ Φ(A)k (Φ(B ∗ B) − Φ(B)∗ Φ(B))
and hence (4.5) holds.
Theorem 4.4. Let a positive linear map Φ : A → B be a unital left multiplier.
If M1 , m1 , M2 , m2 ∈ C and A, B ∈ A satisfy the following conditions:
Re(M1 − A)∗ (A − m1 ) ≥ 0
and
Re(M2 − B)∗ (B − m2 ) ≥ 0,
then
|Φ(A∗ B) − Φ(A)∗ Φ(B)| ≤
1
|M1 − m1 | |M2 − m2 |.
4
Proof. By L¨owner–Heinz theorem, we have
|Φ(A∗ B) − Φ(A)∗ Φ(B)|
1
1
≤
Φ(|A|2 ) − |Φ(A)|2
2 Φ(|B|2 ) − |Φ(B)|2 2
1
≤ |M1 − m1 | |M2 − m2 |
4
(by Lemma 4.3)
(by Lemma 4.1) .
5. Ozeki–Izumino–Mori–Seo type inequality. Let a = (a1 , . . . , an ) and b =
(b1 , . . . , bn ) be n-tuples of real numbers satisfying
0 ≤ m1 ≤ ai ≤ M1
and
0 ≤ m 2 ≤ bi ≤ M 2
(i = 1, . . . , n).
Then Ozeki–Izumino–Mori–Seo inequality [12, 22] asserts that
!2
n
n
n
X
X
X
n2
2
(M1 M2 − m1 m2 ) .
≤
a2i
b2i −
ai bi
3
i=1
i=1
i=1
(5.1)
In [12] they also showed the following operator version of (5.1): If A and B are
positive operators in B(H ) such that 0 < m1 ≤ A ≤ M1 and 0 < m2 ≤ B ≤ M2 for
some scalars m1 ≤ M1 and m2 ≤ M2 , then
(A2 x, x)(B 2 x, x) − (A2 ♯ B 2 x, x)2 ≤
1
2
(M1 M2 − m1 m2 )
4γ 2
(5.2)
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M.S. Moslehian, R. Nakamoto, and Y. Seo
m1 m2
for every unit vector x ∈ H, where γ = max{ M
,
}.
1 M2
Based on the Kantorovich inequality for the difference, we present an extension
of Ozeki–Izumino–Mori–Seo inequality (5.2) as follows.
Theorem 5.1. Suppose that Φ : B(H ) → B(K ) is a positive linear map such
that Φ(I) is invertible and Φ(I) ≤ I. Assume that A, B ∈ B(H ) are positive invertible
operators such that 0 < m1 ≤ A ≤ M1 and 0 < m2 ≤ B ≤ M2 for some scalars
m1 ≤ M1 and m2 ≤ M2 . Then
1
1
1
1
1
1
Φ(B 2 ) 2 Φ(A2 )Φ(B 2 ) 2 − |Φ(B 2 )− 2 Φ(A2 ♯B 2 )Φ(B 2 ) 2 |2 ≤
M2
(M1 M2 − m1 m2 )2
× 22
4
m2
(5.3)
and
1
1
Φ(A2 ) 2 Φ(B 2 )Φ(A2 ) 2 − |Φ(A2 )− 2 Φ(A2 ♯B 2 )Φ(A2 ) 2 |2 ≤
(M1 M2 − m1 m2 )2
M2
× 12 .
4
m1
(5.4)
Proof. Define a normalized positive linear map Ψ by
1
1
1
1
Ψ(X) := Φ(A)− 2 Φ(A 2 XA 2 )Φ(A)− 2 .
By using the Kantorovich inequality for the difference, it follows that
Ψ(X 2 ) − Ψ(X)2 ≤
(M − m)2
4
(5.5)
for all 0 < m ≤ X ≤ M with some scalars m ≤ M . As a matter of fact, we have
Ψ(X 2 ) − Ψ(X)2 ≤ Ψ((M + m)X − M m) − Ψ(X)2
2
(M − m)2
M +m
+
= − Ψ(X) −
2
4
2
(M − m)
.
≤
4
1
1
1
If we put X = (A− 2 BA− 2 ) 2 , then due to
r
r
m2
M2
0 < (m =)
≤X≤
(= M )
M1
m1
we deduce from (5.5) that
− 12
Φ(A)
− 12
Φ(B)Φ(A)
− 21
− Φ(A)
− 21
Φ(A♯B)Φ(A)
2
√
√
( M1 M2 − m1 m2 )2
≤
.
4M1 m1
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A Diaz–Metcalf Type Inequality for Positive Linear Maps
189
Pre- and post-multiplying both sides by Φ(A), we obtain
1
2
1
2
− 21
Φ(A) Φ(B)Φ(A) − |Φ(A)
√
√
( M1 M2 − m1 m2 )2
Φ(A♯B)Φ(A) | ≤
Φ(A)2
4M1 m1
√
√
( M1 M2 − m1 m2 )2
M1
≤
×
,
4
m1
1
2
2
since 0 ≤ Φ(A)2 ≤ M12 . Replacing A and B by A2 and B 2 respectively, we have the
desired inequality (5.4). Similarly, one can obtain (5.3).
Remark 5.2. If Φ is a vector state in (5.3) and (5.4), then we get Ozeki–Izumino–
Mori–Seo inequality (5.2).
Acknowledgement. The authors would like to sincerely thank the anonymous
referee for carefully reading the paper and for valuable comments.
REFERENCES
[1] Lj. Arambaˇsi´
c, D. Baki´
c and M.S. Moslehian. A treatment of the Cauchy–Schwarz inequality
in C ∗ -modules. J. Math. Anal. Appl., to appear, doi:10.1016/j.jmaa.2011.02.062.
[2] P.S. Bullen. A Dictionary of Inequalities. Pitman Monographs and Surveys in Pure and Applied
Mathematics, 97, Longman, Harlow, 1998.
[3] M.-D. Choi. Some assorted inequalities for positive linear maps on C ∗ -algebras. J. Operator
Theory, 4:271–285, 1980.
[4] J.B. Diaz and F.T. Metcalf. Stronger forms of a class of inequalities of G. P´
olya-G. Szeg¨
o and
L.V. Kantorovich. Bull. Amer. Math. Soc., 69:415–418, 1963.
[5] S.S. Dragomir. A survey on Cauchy–Bunyakovsky–Schwarz type discrete inequalities. JIPAM
J. Inequal. Pure Appl. Math., 4, 2003, Article 63.
[6] S.S. Dragomir. A counterpart of Schwarz inequality in inner product spaces. RGMIA Res.
Rep. Coll., 6, 2003, Article 18.
[7] S.S. Dragomir. Advances in Inequalities of the Schwarz, Triangle and Heisenberg Type in Inner
Product Spaces. Nova Science Publishers, Inc., New York, 2007.
[8] D.E. Evans and J.T. Lewis, Dilations of Irreversible Evolutions in Algebraic Quantum Theory.
Comm. Dublin Inst. Adv. Studies. Ser. A, no. 24, 1977.
[9] T. Furuta, J.M. Hot, J.E. Peˇ
cari´
c and Y. Seo. Mond–Peˇ
cari´
c Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space. Monographs in
Inequalities 1, Element, Zagreb, 2005.
[10] W. Greub and W. Rheinboldt. On a generalization of an inequality of L.V. Kantorovich. Proc.
Amer. Math. Soc., 10:407–415, 1959.
[11] D. Iliˇsevi´
c and S. Varoˇsanec. On the Cauchy-Schwarz inequality and its reverse in semi-inner
product C ∗ -modules. Banach J. Math. Anal., 1:78–84, 2007.
[12] S. Izumino, H. Mori and Y. Seo. On Ozeki’s inequality. J. Inequal. Appl., 2:235–253, 1998.
[13] M. Joit¸a. On the Cauchy–Schwarz inequality in C ∗ -algebras. Math. Rep. (Bucur.), 3(53):243–
246, 2001.
[14] M.S. Klamkin and R.G. Mclenaghan. An ellipse inequality. Math. Mag., 50:261–263, 1977.
[15] E.C. Lance. Hilbert C ∗ -Modules. London Math. Soc. Lecture Note Series 210, Cambridge Univ.
Press, 1995.
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 22, pp. 179-190, March 2011
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190
M.S. Moslehian, R. Nakamoto, and Y. Seo
[16] E.-Y. Lee. A matrix reverse Cauchy-Schwarz inequality. Linear Algebra Appl., 430:805–810,
2009.
[17] M.S. Moslehian and L.-E. Persson. Reverse Cauchy-Schwarz inequalities for positive C ∗ -valued
sesquilinear forms. Math. Inequal. Appl., 12:701–709, 2009.
[18] M.S. Moslehian and R. Raji´
c. Gr¨
uss inequality for n-positive linear maps. Linear Alg. Appl.,
433:1555–1560, 2010.
[19] G.J. Murphy, C ∗ -algebras and Operator Theory. Academic Press, Boston, 1990.
[20] C.P. Niculescu. Converses of the Cauchy–Schwarz inequality in the C ∗ -framework. An. Univ.
Craiova Ser. Mat. Inform., 26:22–28, 1999.
[21] M. Niezgoda. Accretive operators and Cassels inequality. Linear Algebra Appl., 433:136–142,
2010.
[22] N. Ozeki. On the estimation of the inequalities by the maximum, or minimum values (in
Japanese). J. College Arts Sci. Chiba Univ., 5:199–203, 1968.
[23] G. P´
olya and G. Szeg¨
o. Aufgaben und Lehrs¨
atze aus der Analysis, Vol. 1, Berlin, 213–214,
1925.
[24] O. Shisha and B. Mond. Bounds on differences of means. Inequalities I, 293–308, 1967.
[25] G.S. Watson. Serial correlation in regression analysis. I. Biometrika, 42:327–342, 1955.
A publication of the International Linear Algebra Society
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ELA
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A DIAZ–METCALF TYPE INEQUALITY FOR POSITIVE LINEAR
MAPS AND ITS APPLICATIONS∗
MOHAMMAD SAL MOSLEHIAN† , RITSUO NAKAMOTO‡ , AND YUKI SEO§
Abstract. We present a Diaz–Metcalf type operator inequality as a reverse Cauchy–Schwarz
inequality and then apply it to get some operator versions of P´
olya–Szeg¨
o’s, Greub–Rheinboldt’s,
Kantorovich’s, Shisha–Mond’s, Schweitzer’s, Cassels’ and Klamkin–McLenaghan’s inequalities via
a unified approach. We also give some operator Gr¨
uss type inequalities and an operator Ozeki–
Izumino–Mori–Seo type inequality. Several applications are included as well.
Key words. Diaz–Metcalf type inequality, Reverse Cauchy–Schwarz inequality, Positive map,
Ozeki–Izumino–Mori–Seo inequality, Operator inequality.
AMS subject classifications. 46L08, 26D15, 46L05, 47A30, 47A63.
1. Introduction. The Cauchy–Schwarz inequality plays an essential role in
mathematical analysis and its applications. In a semi-inner product space (H , h·, ·i)
the Cauchy–Schwarz inequality reads as follows
|hx, yi| ≤ hx, xi1/2 hy, yi1/2
(x, y ∈ H ).
There are interesting generalizations of the Cauchy–Schwarz inequality in various
frameworks, e.g., finite sums, integrals, isotone functionals, inner product spaces, C ∗ algebras and Hilbert C ∗ -modules; see [5, 6, 7, 9, 11, 13, 17, 20] and references therein.
There are several reverses of the Cauchy–Schwarz inequality in the literature: Diaz–
Metcalf’s, P´olya–Szeg¨
o’s, Greub–Rheinboldt’s, Kantorovich’s, Shisha–Mond’s, Ozeki–
Izumino–Mori–Seo’s, Schweitzer’s, Cassels’ and Klamkin–McLenaghan’s inequalities.
Inspired by the work of J.B. Diaz and F.T. Metcalf [4], we present several reverse
Cauchy–Schwarz type inequalities for positive linear maps. We give a unified treatment of some reverse inequalities of the classical Cauchy–Schwarz type for positive
∗ Received by the editors on September 29, 2010. Accepted for publication on February 19, 2011
Handling Editor: Bit-Shun Tam.
† Department of Pure Mathematics, Centre of Excellence in Analysis on Algebraic Structures
(CEAAS), Ferdowsi University of Mashhad, P.O. Box 1159, Mashhad 91775, Iran (moslehian@
ferdowsi.um.ac.ir, [email protected]). Supported by a grant from Ferdowsi University of Mashhad
(No. MP89144MOS).
‡ Faculty of Engineering, Ibaraki University, Hitachi, Ibaraki 316-0033, Japan (r-naka@net1.
jway.ne.jp).
§ Faculty of Engineering, Shibaura Institute of Technology, 307 Fukasaku, Minuma-ku, Saitamacity, Saitama 337-8570, Japan ([email protected]).
179
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180
M.S. Moslehian, R. Nakamoto, and Y. Seo
linear maps.
Throughout the paper B(H ) stands for the algebra of all bounded linear operators
acting on a Hilbert space H . We simply denote by α the scalar multiple αI of the
identity operator I ∈ B(H ). For self-adjoint operators A, B the partially ordered
relation B ≤ A means that hBξ, ξi ≤ hAξ, ξi for all ξ ∈ H . In particular, if 0 ≤ A,
then A is called positive. If A is a positive invertible operator, then we write 0 < A. A
linear map Φ : A → B between C ∗ -algebras is said to be positive if Φ(A) is positive
whenever A is. We say that Φ is unital if Φ preserves the identity. The reader is
referred to [9, 19] for undefined notations and terminologies.
2. Operator Diaz–Metcalf type inequality. We start this section with our
main result. Recall that the geometric operator mean A ♯ B for positive operators
A, B ∈ B(H ) is defined by
1
21 1
1
1
A ♯ B = A 2 A− 2 BA− 2
A2
if 0 < A.
Theorem 2.1. Let A, B ∈ B(H ) be positive invertible operators and Φ :
B(H ) → B(K ) be a positive linear map.
(i) If m2 A ≤ B ≤ M 2 A for some positive real numbers m < M , then the
following inequalities hold:
• Operator Diaz–Metcalf inequality of first type
M mΦ(A) + Φ(B) ≤ (M + m)Φ(A♯B) ;
• Operator Cassels inequality
M +m
Φ(A)♯Φ(B) ≤ √
Φ(A♯B) ;
2 Mm
• Operator Klamkin–McLenaghan inequality
Φ(A♯B)
−1
2
Φ(B)Φ(A♯B)
−1
2
√
√
1
1
− Φ(A♯B) 2 Φ(A)−1 Φ(A♯B) 2 ≤ ( M − m)2 ;
• Operator Kantorovich inequality
Φ(A)♯Φ(A−1 ) ≤
M 2 + m2
.
2M m
(ii) If m21 ≤ A ≤ M12 and m22 ≤ B ≤ M22 for some positive real numbers
m1 < M1 and m2 < M2 , then the following inequalities hold:
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A Diaz–Metcalf Type Inequality for Positive Linear Maps
• Operator Diaz–Metcalf inequality of second type
m2
M2
M 2 m2
Φ(A) + Φ(B) ≤
+
Φ(A♯B) ;
M 1 m1
m1
M1
• Operator P´
olya–Szeg¨
o inequality
!
r
r
M1 M2
m1 m2
1
Φ(A♯B) ;
+
Φ(A)♯Φ(B) ≤
2
m1 m 2
M 1 M2
• Operator Shisha–Mond inequality
Φ(A♯B)
−1
2
Φ(B)Φ(A♯B)
−1
2
1
1
− Φ(A♯B) 2 Φ(A)−1 Φ(A♯B) 2
!2
r
r
M2
m2
≤
−
;
m1
M1
• Operator Gr¨
uss type inequality
2
√
√
√
M1 M2
M 1 M 2 − m 1 m2
M1 M2
.
min
,
Φ(A)♯Φ(B) − Φ(A♯B) ≤
√
2 m1 m2
m 1 m2
Proof.
(i) If m2 A ≤ B ≤ M 2 A for some positive real numbers m < M , then
−1
−1
m ≤ A 2 BA 2 ≤ M 2 .
2
(ii) If m21 ≤ A ≤ M12 and m22 ≤ B ≤ M22 for some positive real numbers
m1 < M1 and m2 < M2 , then
m2 =
−1
−1
m22
M22
2 BA 2 ≤
≤
A
= M2 .
M12
m21
(2.1)
In any case we then have
1/2 −1
1/2
−1
−1
−1
A 2 BA 2
M − A 2 BA 2
− m ≥ 0,
whence
Mm + A
Hence
−1
2
BA
−1
2
12
−1
−1
≤ (M + m) A 2 BA 2
.
−1
21
−1
M mA + B ≤ (M + m)A1/2 A 2 BA 2
A1/2 = (M + m)A♯B .
(2.2)
M mΦ(A) + Φ(B) ≤ (M + m)Φ(A♯B) .
(2.3)
Since Φ is a positive linear map, (2.2) yields the operator Diaz–Metcalf inequality of
first type as follows:
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M.S. Moslehian, R. Nakamoto, and Y. Seo
In the case when (ii) holds we get the following, which is called the operator Diaz–
Metcalf inequality of second type:
M 2 m2
m2
M2
Φ(A) + Φ(B) ≤
+
Φ(A♯B) .
M 1 m1
m1
M1
Following the strategy of [21], we apply the operator geometric–arithmetic inequality
to M mΦ(A) and Φ(B) to get:
√
1
M m(Φ(A)♯Φ(B)) = (M mΦ(A))♯Φ(B) ≤ (M mΦ(A) + Φ(B)) .
2
(2.4)
It follows from (2.3) and (2.4) that
M +m
Φ(A♯B) ,
Φ(A)♯Φ(B) ≤ √
2 Mm
which is said to be the operator Cassels inequality under the assumption (i); see also
[16]. Under the case (ii) we can represent it as the following inequality being called
the operator P´
olya–Szeg¨
o inequality or the operator Greub–Rheinboldt inequality:
!
r
r
1
M1 M2
m1 m 2
Φ(A)♯Φ(B) ≤
+
Φ(A♯B) .
(2.5)
2
m1 m 2
M 1 M2
It follows from (2.5) that
!
!
r
M1 M2
m 1 m2
− 1 Φ(A♯B)
+
m1 m2
M1 M2
2
√
M 1 M 2 − m 1 m2
√
Φ(A♯B) .
√
2 m 1 m2 M 1 M 2
1
Φ(A)♯Φ(B) − Φ(A♯B) ≤
2
√
=
r
(2.6)
It follows from (2.1) that
so
1/2
−1
−1
m2
M2
A1/2 ≤
A ≤ A1/2 A 2 BA 2
A,
M1
m1
M 2 M2
m21 m2
≤ A♯B ≤ 1
.
M1
m1
Now, (2.6) and (2.7) yield that
Φ(A)♯Φ(B) − Φ(A♯B) ≤
√
2
√
M1 M2 − m1 m2 M12 M2
√
.
√
m1
2 m1 m2 M 1 M 2
An easy symmetric argument then follows that
2
√
√
√
M1 M2
M 1 M 2 − m 1 m2
M 1 M2
Φ(A)♯Φ(B) − Φ(A♯B) ≤
,
min
,
√
2 m1 m2
m 1 m2
(2.7)
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A Diaz–Metcalf Type Inequality for Positive Linear Maps
presenting a Gr¨
uss type inequality.
If A is invertible and Φ is unital and m21 = m2 ≤ A ≤ M 2 = M12 , then by
putting m22 = 1/M 2 ≤ B = A−1 ≤ 1/m2 = M22 in (2.5) we get the following operator
Kantorovich inequality:
Φ(A)♯Φ(A−1 ) ≤
M 2 + m2
.
2M m
It follows from (2.3) that
Φ(A♯B)
−1
2
Φ(B)Φ(A♯B)
−1
2
−1
1
1
− Φ(A♯B) 2 Φ(A)−1 Φ(A♯B) 2
1
−1
1
≤ M + m − M mΦ(A♯B) 2 Φ(A)Φ(A♯B) 2 − Φ(A♯B) 2 Φ(A)−1 Φ(A♯B) 2
√
1/2
√
−1
−1
≤ M + m − 2 Mm −
M m Φ(A♯B) 2 Φ(A)Φ(A♯B) 2
1/2 2
1
1
− Φ(A♯B) 2 Φ(A)−1 Φ(A♯B) 2
√
√
≤ ( M − m)2 ,
(2.8)
that is, an operator Klakmin–Mclenaghan inequality when (i) holds. Under (ii), we
get the following operator Shisha–Szeg¨
o inequality from (2.8):
!2
r
r
−1
−1
1
1
M
m
2
2
−
.
Φ(A♯B) 2 Φ(B)Φ(A♯B) 2 − Φ(A♯B) 2 Φ(A)−1 Φ(A♯B) 2 ≤
m1
M1
3. Applications. If (a1 , . . . , an ) and (b1 , . . . , bn ) are n-tuples of real numbers
with 0 < m1 ≤ ai ≤ M1 (1 ≤ i ≤ n), 0 < m2 ≤ bi ≤ M2 (1 ≤ i ≤ n), we can
consider the positive linear map Φ(T ) = hT x, xi on B(Cn ) = Mn (C) and let A =
diag(a21 , . . . , a2n ), B = diag(b21 , . . . , b2n ) and x = (1, . . . , 1)t in the operator inequalities
above to get the following classical inequalities:
• Diaz–Metcalf inequality [4]
n
X
k=1
b2k
n
n
m2 X
M2
m2 M 2 X 2
ak ≤
+
+
ak bk .
m1 M 1
m1
M1
k=1
k=1
• P´olya–Szeg¨
o inequality [23]
Pn
Pn
2
2
1
k=1 ak
k=1 bk
≤
Pn
2
4
( k=1 ak bk )
r
M1 M2
+
m1 m 2
r
!2
;
!2
;
m1 m2
M1 M2
• Shisha–Mond inequality [24]
Pn
Pn
a2
k=1 ak bk
Pnk=1 k − P
n
2 ≤
a
b
k
k
k=1
k=1 bk
r
M1
−
m2
r
m1
M2
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M.S. Moslehian, R. Nakamoto, and Y. Seo
• A Gr¨
uss type inequality
n
X
a2k
k=1
!1/2
n
X
k=1
b2k
!1/2
−
n
X
ak bk
k=1
2
√
√
√
M1 M2
M 1 M 2 − m 1 m2
M1 M2
min
,
.
≤
√
2 m1 m 2
m1 m 2
Using the same argument with a positive n-tuple (a1 , . . . , an ) of real numbers with
0 < m ≤ ai ≤ M (1 ≤ i ≤ n), x = √1n (1, . . . , 1)t , we get from Kantorovich inequality
that
• Schweitzer inequality [2]
n
1X 2
a
n i=1 i
!
n
1 X −2
a
n i=1 i
!
≤
(M 2 + m2 )2
.
4M 2 m2
If (a1 , . . . , an ) and (b1 , . . . , bn ) are n-tuples of real numbers with 0 < m ≤ ai /bi ≤
M (1 ≤ i ≤ n), we can consider the positive linear map Φ(T ) = hT x, xi on B(Cn ) =
√
√
Mn (C) and let A = diag(a21 , . . . , a2n ), B = diag(b21 , . . . , b2n ) and x = ( w1 , . . . , wn )t
based on the weight w
¯ = (w1 , . . . , wn ), in the operator inequalities above to get the
following classical inequalities:
• Cassels inequality [25]
Pn
Pn
2
wk a2k k=1 wk b2k
(M + m)
;
≤
Pn
2
4mM
( k=1 wk ak bk )
k=1
• Klamkin–McLenaghan inequality [14]
n
X
k=1
wk a2k
n
X
k=1
wk b2k
−
n
X
wk ak bk
k=1
!2
≤
n
n
√
X
√ 2 X
wk ak bk
M− m
wk a2k .
k=1
k=1
Using the same argument, we obtain a weighted form of the P´olya–Szeg¨o inequality as follows:
• Grueb–Rheinboldt inequality [10]
Pn
2
wk a2k k=1 wk b2k
(M1 M2 + m1 m2 )
≤
.
Pn
2
4m1 m2 M1 M2
( k=1 wk ak bk )
Pn
k=1
One can assert the integral versions of discrete results above by considering
L2 (X, µ), where (X, µ) is a probability space, as a Hilbert space via hh1 , h2 i =
R
h h dµ, multiplication operators A, B ∈ B(L2 (X, µ))) defined by A(h) = f 2 h
X 1 2
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A Diaz–Metcalf Type Inequality for Positive Linear Maps
and B(h) = g 2 h for bounded f, g ∈ L2 (X, µ) and a positive linear map Φ by
R
Φ(T ) = X T (1)dµ on B(L2 (X, µ))). For instance, let us state integral versions of the
Cassels and Klamkin–McLenaghan inequalities. These two inequalities are obtained,
first for bounded positive functions f, g ∈ L2 (X, µ) and next for general positive
functions f, g ∈ L2 (X, µ) as the limits of sequences of bounded positive functions.
Corollary 3.1. Let (X, µ) be a probability space and f, g ∈ L2 (X, µ) with
0 ≤ mg ≤ f ≤ M g for some scalars 0 < m < M . Then
Z
f 2 dµ
X
Z
X
g 2 dµ ≤
(M + m)2
4M m
Z
X
2
f gdµ
and
Z
2
f dµ
X
Z
2
X
g dµ −
Z
X
2
Z
Z
√
√ 2
M− m
f gdµ
f 2 dµ .
f gdµ ≤
X
X
Pn
Considering the positive linear functional Φ(R) = i=1 hRξi , ξi i on B(H ), where
ξ1 , . . . , ξn ∈ H , we get the following versions of the Diaz–Metcalf and P´olya–Szeg¨o
inequalities in a Hilbert space.
Corollary 3.2. Let H be a Hilbert space, let ξ1 , . . . , ξn ∈ H and let T, S ∈
B(H ) be positive operators satisfying 0 < m1 ≤ T ≤ M1 and 0 < m2 ≤ S ≤ M2 .
Then
n
n
n
X
M 2 m2 X
m2 X
M2
kT ξi k2 +
kSξi k2 ≤
+
k(T 2 ♯S 2 )1/2 ξi k2
M1 m1 i=1
m
M
1
1
i=1
i=1
and
n
X
i=1
2
kT ξi k
!1/2
n
X
i=1
1
≤
2
2
kSξi k
r
!1/2
M1 M2
+
m 1 m2
r
m1 m2
M1 M2
!
n
X
i=1
k(T 2 ♯S 2 )1/2 ξi k2 .
4. A Gr¨
uss type inequality. In this section we obtain another Gr¨
uss type
∗
∗
inequality, see also [18]. Let A be a C -algebra and let B be a C -subalgebra of
A . Following [1], a positive linear map Φ : A → B is called a left multiplier if
Φ(XY ) = Φ(X)Y for every X ∈ A , Y ∈ B.
The following lemma is interesting on its own right.
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Lemma 4.1. Let Φ be a unital positive linear map on A , A ∈ A and M, m be
complex numbers such that
Re ((M − A)∗ (A − m)) ≥ 0 .
(4.1)
Then
2
Φ(|A|2 ) − |Φ(A)| ≤
1
|M − m|2 .
4
Proof. For any complex number c ∈ C, we have
Φ(|A|2 ) − |Φ(A)|2 = Φ(|A − c|2 ) − |Φ(A − c)|2 .
(4.2)
Since for any T ∈ A the operator equality
2
M + m
1
2
= Re ((M − T )(T − m)∗ )
|M − m| − T −
4
2
holds, the condition (4.1) implies that
2 !
M
+
m
1
Φ A −
≤ |M − m|2 .
2
4
(4.3)
Therefore, it follows from (4.2) and (4.3) that
Φ(|A|2 ) − |Φ(A)|2 ≤ Φ(|A −
≤
M +m 2
| )
2
1
|M − m|2 .
4
Remark 4.2. If (i) Φ is a unital positive linear map and A is a normal operator
or (ii) Φ is a 2-positive linear map and A is an arbitrary operator, then it follows from
[3] that
2
0 ≤ Φ(|A|2 ) − |Φ(A)| .
(4.4)
Condition (4.4) is stronger than positivity and weaker than 2-positivity; see [8]. Another class of positive linear maps satisfying (4.4) are left multipliers, cf. [1, Corollary
2.4].
Lemma 4.3. Let a positive linear map Φ : A → B be a unital left multiplier.
Then
2
|Φ(A∗ B) − Φ(A)∗ Φ(B)| ≤
Φ(|A|2 ) − |Φ(A)|2
Φ(|B|2 ) − |Φ(B)|2
(4.5)
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A Diaz–Metcalf Type Inequality for Positive Linear Maps
Proof. If we put [X, Y ] := Φ(X ∗ Y ) − Φ(X)∗ Φ(Y ), then A is a right pre-inner
product C ∗ -module over B, since Φ(X ∗ Y ) is a right pre-inner product B-module, see
[1, Corollary 2.4]. It follows from the Cauchy–Schwarz inequality in pre-inner product
C ∗ -modules (see [15, Proposition 1.1]) that
2
|Φ(A∗ B) − Φ(A)∗ Φ(B)| = [B, A][A, B]
≤ k[A, A]k[B, B]
= kΦ(A∗ A) − Φ(A)∗ Φ(A)k (Φ(B ∗ B) − Φ(B)∗ Φ(B))
and hence (4.5) holds.
Theorem 4.4. Let a positive linear map Φ : A → B be a unital left multiplier.
If M1 , m1 , M2 , m2 ∈ C and A, B ∈ A satisfy the following conditions:
Re(M1 − A)∗ (A − m1 ) ≥ 0
and
Re(M2 − B)∗ (B − m2 ) ≥ 0,
then
|Φ(A∗ B) − Φ(A)∗ Φ(B)| ≤
1
|M1 − m1 | |M2 − m2 |.
4
Proof. By L¨owner–Heinz theorem, we have
|Φ(A∗ B) − Φ(A)∗ Φ(B)|
1
1
≤
Φ(|A|2 ) − |Φ(A)|2
2 Φ(|B|2 ) − |Φ(B)|2 2
1
≤ |M1 − m1 | |M2 − m2 |
4
(by Lemma 4.3)
(by Lemma 4.1) .
5. Ozeki–Izumino–Mori–Seo type inequality. Let a = (a1 , . . . , an ) and b =
(b1 , . . . , bn ) be n-tuples of real numbers satisfying
0 ≤ m1 ≤ ai ≤ M1
and
0 ≤ m 2 ≤ bi ≤ M 2
(i = 1, . . . , n).
Then Ozeki–Izumino–Mori–Seo inequality [12, 22] asserts that
!2
n
n
n
X
X
X
n2
2
(M1 M2 − m1 m2 ) .
≤
a2i
b2i −
ai bi
3
i=1
i=1
i=1
(5.1)
In [12] they also showed the following operator version of (5.1): If A and B are
positive operators in B(H ) such that 0 < m1 ≤ A ≤ M1 and 0 < m2 ≤ B ≤ M2 for
some scalars m1 ≤ M1 and m2 ≤ M2 , then
(A2 x, x)(B 2 x, x) − (A2 ♯ B 2 x, x)2 ≤
1
2
(M1 M2 − m1 m2 )
4γ 2
(5.2)
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M.S. Moslehian, R. Nakamoto, and Y. Seo
m1 m2
for every unit vector x ∈ H, where γ = max{ M
,
}.
1 M2
Based on the Kantorovich inequality for the difference, we present an extension
of Ozeki–Izumino–Mori–Seo inequality (5.2) as follows.
Theorem 5.1. Suppose that Φ : B(H ) → B(K ) is a positive linear map such
that Φ(I) is invertible and Φ(I) ≤ I. Assume that A, B ∈ B(H ) are positive invertible
operators such that 0 < m1 ≤ A ≤ M1 and 0 < m2 ≤ B ≤ M2 for some scalars
m1 ≤ M1 and m2 ≤ M2 . Then
1
1
1
1
1
1
Φ(B 2 ) 2 Φ(A2 )Φ(B 2 ) 2 − |Φ(B 2 )− 2 Φ(A2 ♯B 2 )Φ(B 2 ) 2 |2 ≤
M2
(M1 M2 − m1 m2 )2
× 22
4
m2
(5.3)
and
1
1
Φ(A2 ) 2 Φ(B 2 )Φ(A2 ) 2 − |Φ(A2 )− 2 Φ(A2 ♯B 2 )Φ(A2 ) 2 |2 ≤
(M1 M2 − m1 m2 )2
M2
× 12 .
4
m1
(5.4)
Proof. Define a normalized positive linear map Ψ by
1
1
1
1
Ψ(X) := Φ(A)− 2 Φ(A 2 XA 2 )Φ(A)− 2 .
By using the Kantorovich inequality for the difference, it follows that
Ψ(X 2 ) − Ψ(X)2 ≤
(M − m)2
4
(5.5)
for all 0 < m ≤ X ≤ M with some scalars m ≤ M . As a matter of fact, we have
Ψ(X 2 ) − Ψ(X)2 ≤ Ψ((M + m)X − M m) − Ψ(X)2
2
(M − m)2
M +m
+
= − Ψ(X) −
2
4
2
(M − m)
.
≤
4
1
1
1
If we put X = (A− 2 BA− 2 ) 2 , then due to
r
r
m2
M2
0 < (m =)
≤X≤
(= M )
M1
m1
we deduce from (5.5) that
− 12
Φ(A)
− 12
Φ(B)Φ(A)
− 21
− Φ(A)
− 21
Φ(A♯B)Φ(A)
2
√
√
( M1 M2 − m1 m2 )2
≤
.
4M1 m1
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A Diaz–Metcalf Type Inequality for Positive Linear Maps
189
Pre- and post-multiplying both sides by Φ(A), we obtain
1
2
1
2
− 21
Φ(A) Φ(B)Φ(A) − |Φ(A)
√
√
( M1 M2 − m1 m2 )2
Φ(A♯B)Φ(A) | ≤
Φ(A)2
4M1 m1
√
√
( M1 M2 − m1 m2 )2
M1
≤
×
,
4
m1
1
2
2
since 0 ≤ Φ(A)2 ≤ M12 . Replacing A and B by A2 and B 2 respectively, we have the
desired inequality (5.4). Similarly, one can obtain (5.3).
Remark 5.2. If Φ is a vector state in (5.3) and (5.4), then we get Ozeki–Izumino–
Mori–Seo inequality (5.2).
Acknowledgement. The authors would like to sincerely thank the anonymous
referee for carefully reading the paper and for valuable comments.
REFERENCES
[1] Lj. Arambaˇsi´
c, D. Baki´
c and M.S. Moslehian. A treatment of the Cauchy–Schwarz inequality
in C ∗ -modules. J. Math. Anal. Appl., to appear, doi:10.1016/j.jmaa.2011.02.062.
[2] P.S. Bullen. A Dictionary of Inequalities. Pitman Monographs and Surveys in Pure and Applied
Mathematics, 97, Longman, Harlow, 1998.
[3] M.-D. Choi. Some assorted inequalities for positive linear maps on C ∗ -algebras. J. Operator
Theory, 4:271–285, 1980.
[4] J.B. Diaz and F.T. Metcalf. Stronger forms of a class of inequalities of G. P´
olya-G. Szeg¨
o and
L.V. Kantorovich. Bull. Amer. Math. Soc., 69:415–418, 1963.
[5] S.S. Dragomir. A survey on Cauchy–Bunyakovsky–Schwarz type discrete inequalities. JIPAM
J. Inequal. Pure Appl. Math., 4, 2003, Article 63.
[6] S.S. Dragomir. A counterpart of Schwarz inequality in inner product spaces. RGMIA Res.
Rep. Coll., 6, 2003, Article 18.
[7] S.S. Dragomir. Advances in Inequalities of the Schwarz, Triangle and Heisenberg Type in Inner
Product Spaces. Nova Science Publishers, Inc., New York, 2007.
[8] D.E. Evans and J.T. Lewis, Dilations of Irreversible Evolutions in Algebraic Quantum Theory.
Comm. Dublin Inst. Adv. Studies. Ser. A, no. 24, 1977.
[9] T. Furuta, J.M. Hot, J.E. Peˇ
cari´
c and Y. Seo. Mond–Peˇ
cari´
c Method in Operator Inequalities. Inequalities for Bounded Selfadjoint Operators on a Hilbert Space. Monographs in
Inequalities 1, Element, Zagreb, 2005.
[10] W. Greub and W. Rheinboldt. On a generalization of an inequality of L.V. Kantorovich. Proc.
Amer. Math. Soc., 10:407–415, 1959.
[11] D. Iliˇsevi´
c and S. Varoˇsanec. On the Cauchy-Schwarz inequality and its reverse in semi-inner
product C ∗ -modules. Banach J. Math. Anal., 1:78–84, 2007.
[12] S. Izumino, H. Mori and Y. Seo. On Ozeki’s inequality. J. Inequal. Appl., 2:235–253, 1998.
[13] M. Joit¸a. On the Cauchy–Schwarz inequality in C ∗ -algebras. Math. Rep. (Bucur.), 3(53):243–
246, 2001.
[14] M.S. Klamkin and R.G. Mclenaghan. An ellipse inequality. Math. Mag., 50:261–263, 1977.
[15] E.C. Lance. Hilbert C ∗ -Modules. London Math. Soc. Lecture Note Series 210, Cambridge Univ.
Press, 1995.
Electronic Journal of Linear Algebra ISSN 1081-3810
A publication of the International Linear Algebra Society
Volume 22, pp. 179-190, March 2011
ELA
http://math.technion.ac.il/iic/ela
190
M.S. Moslehian, R. Nakamoto, and Y. Seo
[16] E.-Y. Lee. A matrix reverse Cauchy-Schwarz inequality. Linear Algebra Appl., 430:805–810,
2009.
[17] M.S. Moslehian and L.-E. Persson. Reverse Cauchy-Schwarz inequalities for positive C ∗ -valued
sesquilinear forms. Math. Inequal. Appl., 12:701–709, 2009.
[18] M.S. Moslehian and R. Raji´
c. Gr¨
uss inequality for n-positive linear maps. Linear Alg. Appl.,
433:1555–1560, 2010.
[19] G.J. Murphy, C ∗ -algebras and Operator Theory. Academic Press, Boston, 1990.
[20] C.P. Niculescu. Converses of the Cauchy–Schwarz inequality in the C ∗ -framework. An. Univ.
Craiova Ser. Mat. Inform., 26:22–28, 1999.
[21] M. Niezgoda. Accretive operators and Cassels inequality. Linear Algebra Appl., 433:136–142,
2010.
[22] N. Ozeki. On the estimation of the inequalities by the maximum, or minimum values (in
Japanese). J. College Arts Sci. Chiba Univ., 5:199–203, 1968.
[23] G. P´
olya and G. Szeg¨
o. Aufgaben und Lehrs¨
atze aus der Analysis, Vol. 1, Berlin, 213–214,
1925.
[24] O. Shisha and B. Mond. Bounds on differences of means. Inequalities I, 293–308, 1967.
[25] G.S. Watson. Serial correlation in regression analysis. I. Biometrika, 42:327–342, 1955.