Xue and Hu Journal of Inequalities and Applications (2016) 2016:27 DOI 10.1186/s13660-016-0976-5
RESEARCH
Open Access
Some generalizations of operator inequalities for positive linear maps Jianming Xue1* and Xingkai Hu2 *
Correspondence:
[email protected] 1 Oxbridge College, Kunming University of Science and Technology, Kunming, Yunnan 650106, P.R. China Full list of author information is available at the end of the article
Abstract In this paper, we generalize some operator inequalities for positive linear maps due to Lin (Stud. Math. 215:187-194, 2013) and Zhang (Banach J. Math. Anal. 9:166-172, 2015). MSC: 47A63; 47A30 Keywords: operator inequalities; Kantorovich inequalities; positive linear maps
1 Introduction Throughout this paper, let M, M , m, m be scalars, I be the identity operator, and B (H) be the set of all bounded linear operators on a Hilbert space (H, ·, ·). The operator norm is denoted by · . We write A ≥ if the operator A is positive. If A – B ≥ , then we say that A ≥ B. For A, B > , we use the following notation: A ∇μ B = ( – μ)A + μB, A μ B = A (A– BA– )μ A , where ≤ μ ≤ . When μ = we write A ∇ B and A B for brevity for A ∇ B and A B, respectively; see Kubo and Ando []. A linear map is positive if (A) ≥ whenever A ≥ . It is said to be unital if (I) = I. We say that is -positive if whenever the × operator matrix BA∗ CB is positive, then (A) (B) so is (B∗ ) (C) . Let < m ≤ A, B ≤ M and be positive unital linear map. Lin [], Theorem ., proved the following reversed operator AM-GM inequalities:
A+B A+B
≤ K (h) (A B),
(.)
≤ K (h) (A) (B) ,
(.)
with h = M is the Kantorovich constant. where K(h) = (h+) h m Can the inequalities (.) and (.) be improved? Lin [], Conjecture ., conjectured
h h–
that the constant K(h) can be replaced by the Specht ratio S(h) =
in (.) and (.),
e log h h–
which remains as an open question. Zhang [], Theorem ., generalized (.) and (.) when p ≥ : p
A+B
≤
(K(M + m ))p p (A B), Mp mp
(.)
© 2016 Xue and Hu. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Xue and Hu Journal of Inequalities and Applications (2016) 2016:27
p
A+B
≤
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p (K(M + m ))p (A) (B) . p p M m
(.)
We will present some operator inequalities which are generalizations of (.), (.), (.), and (.) in the next section. Bhatia and Davis [] proved that if < m ≤ A ≤ M and X and Y are two partial isometries on H whose final spaces are orthogonal to each other. Then for every -positive unital linear map , – X ∗ AY Y ∗ AY Y ∗ AX ≤
M–m M+m
X ∗ AX .
(.)
Lin [], Conjecture ., conjectured that the following inequality could be true: ∗ ∗ – ∗ ∗ – X AY Y AY Y AX X AX ≤
M–m M+m
.
Recently, Fu and He [], Theorem , proved ∗ ∗ – ∗ ∗ – X AY Y AY Y AX X AX ≤
M–m M+m
M+
m
.
(.)
We will get a stronger result than (.).
2 Main results We begin this section with the following lemmas. Lemma [] Let A, B > . Then the following norm inequality holds: AB ≤
A + B .
(.)
Lemma [] Let A > . Then for every positive unital linear map , A– ≥ – (A).
(.)
Lemma [] Let A, B > . Then, for ≤ r < ∞, r A + Br ≤ (A + B)r .
(.)
Lemma ([], Theorem ) Suppose that two operators A, B and positive real numbers m, m , M, M satisfy either of the following conditions: () < m ≤ A ≤ m < M ≤ B ≤ M, () < m ≤ B ≤ m < M ≤ A ≤ M. Then A ∇μ B ≥ K r h A μ B for all μ ∈ [, ], where r = min(μ, – μ) and h =
M . m
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Theorem Let < m ≤ A ≤ m < M ≤ B ≤ M. Then A+B + MmK h (A B)– ≤ M + m, where K(h ) =
(h +) h
with h =
(.)
M . m
Proof It is easy to see that (M – A)(m – A)A– ≤ , then Mm
A– A M + m + ≤ .
Similarly,
Mm
B– B M + m + ≤ .
Summing up the above two inequalities, we get A– + B– A+B + Mm ≤ M + m. By (A B)– = A– B– and Lemma , we have A+B A+B + MmK h (A B)– = + MmK h A– B– A– + B– A+B + Mm ≤ M + m.
≤
This completes the proof.
Theorem Let < m ≤ A ≤ m < M ≤ B ≤ M. Then for every positive unital linear map ,
A+B
≤
K (h) (A B) K(h )
(.)
≤
K (h) (A) (B) , K(h )
(.)
and
A+B
where K(h) =
(h+) , h
K(h ) =
(h +) , h
h=
M , m
and h =
M . m
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Proof The inequality (.) is equivalent to A + B – (A B) ≤ K(h) . K (h )
(.)
Compute A + B MmK h – (A B) A+B – + MmK h (A B) ≤ (by (.)) A+B – + MmK h (A B) (by (.)) ≤ A+B – + MmK h (A B) = ≤ (M + m) (by (.)) = (M + m) . That is, A + B – (A B) ≤ (M + m) = K(h) . MmK (h ) K (h ) Thus, (.) holds. The proof of (.) is similar, we omit the details. This completes the proof. Remark Because of and (.), respectively.
K (h) K(h )
< K (h), inequalities (.) and (.) are refinements of (.)
Theorem Let < m ≤ A ≤ m < M ≤ B ≤ M and ≤ p < ∞. Then for every positive unital linear map , p
A+B
≤
K (h)(M + m ) p p (A B) K(h )M m
(.)
≤
p K (h)(M + m ) p (A) (B) , K(h )M m
(.)
and p
A+B
where K(h) =
(h+) , h
K(h ) =
(h +) , h
h=
M , m
and h =
M . m
Proof By the operator reverse monotonicity of inequality (.), we have – (A B) ≤ L – where L =
K(h)
K (h )
.
A+B ,
(.)
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Compute p A+B p p –p M m (A B) p p M m p A+B –p ≤ L (A B) + (by (.)) L p A+B M m – (by (.)) (A B) ≤ L + L p A+B – A + B ≤ m L + LM (by (.)) p ≤ L M + m (by [], (.)). That is, p p A+B L(M + m ) p K (h)(M + m ) –p (A B) = . ≤ Mm K(h )M m Thus, (.) holds. By inequality (.), the proof of (.) is similar, we omit the details. This completes the proof. Remark Since K(h ) > , inequalities (.) and (.) are sharper than (.) and (.), respectively. Theorem Let < m ≤ A ≤ M and let X, Y be two isometries on H whose final spaces are orthogonal to each other. Then for every -positive unital linear map , ∗ ∗ – ∗ ∗ – (M – m) X AY Y AY Y AX X AX ≤ . Mm Proof Since X is isometric and < m ≤ A ≤ M, m ≤ (X ∗ AX) ≤ M and (X ∗ AX)– ≤ m . Compute
(.) M
≤
– – M–m Mm X ∗ AY Y ∗ AY Y ∗ AX X ∗ AX M+m ∗ – ∗ ∗ – ∗ M–m (by (.)) ≤ X AY Y AY Y AX + Mm X AX M+m ∗ – M–m ∗ M–m (by (.)) X AX + Mm X AX ≤ M + m M+m M–m ≤ (M + m) . M+m
Hence, ∗ ∗ – ∗ ∗ – (M – m) X AY Y AY Y AX X AX ≤ . Mm This completes the proof.
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Remark Since < m ≤ M,
M–m M+m
M+ m
≥
M–m M+m
M ≥ m
M–m M+m
(M + m) (M – m) = . Mm Mm
Thus, (.) is tighter than (.).
Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript. Author details 1 Oxbridge College, Kunming University of Science and Technology, Kunming, Yunnan 650106, P.R. China. 2 Faculty of Science, Kunming University of Science and Technology, Kunming, Yunnan 650500, P.R. China. Acknowledgements The authors wish to express their heartfelt thanks to the referees for their detailed and helpful suggestions for revising the manuscript. This research was supported by Scientific Research Fund of Yunnan Provincial Education Department (No. 2014C206Y). Received: 9 October 2015 Accepted: 15 January 2016 References 1. Lin, M: Squaring a reverse AM-GM inequality. Stud. Math. 215, 187-194 (2013) 2. Zhang, P: More operator inequalities for positive linear maps. Banach J. Math. Anal. 9, 166-172 (2015) 3. Kubo, F, Ando, T: Means of positive linear operators. Math. Ann. 246, 205-244 (1980) 4. Bhatia, R, Davis, C: More operator versions of the Schwarz inequality. Commun. Math. Phys. 215, 239-244 (2000) 5. Lin, M: On an operator Kantorovich inequality for positive linear maps. J. Math. Anal. Appl. 402, 127-132 (2013) 6. Fu, X, He, C: Some operator inequalities for positive linear maps. Linear Multilinear Algebra 65, 571-577 (2015) 7. Bhatia, R, Kittaneh, F: Notes on matrix arithmetic-geometric mean inequalities. Linear Algebra Appl. 308, 203-211 (2000) 8. Bhatia, R: Positive Definite Matrices. Princeton University Press, Princeton (2007) 9. Ando, T, Zhan, X: Norm inequalities related to operator monotone functions. Math. Ann. 315, 771-780 (1999) 10. Zuo, H, Shi, G, Fujii, M: Refined Young inequality with Kantorovich constant. J. Math. Inequal. 5, 551-556 (2011)
