#### Some means inequalities for positive operators in Hilbert spaces

Liang and Shi Journal of Inequalities and Applications
Some means inequalities for positive operators in Hilbert spaces
Jin Liang
Guanghua Shi
In this paper, we obtain two refinements of the ordering relations among Heinz means with different parameters via the Taylor series of some hyperbolic functions and by the way, we derive new generalizations of Heinz operator inequalities. Moreover, we establish a matrix version of Heinz inequality for the Hilbert-Schmidt norm. Finally, we introduce a weighted multivariate geometric mean and show that the weighted multivariate operator geometric mean possess several attractive properties and means inequalities.
means inequalities; positive linear operators; Hilbert space
1 Introduction
Since Heinz proved a series of useful norm inequalities, which are closely related to the
Cordes inequality and the Furuta inequality, in , many researchers have devoted
themselves to sharping the Heinz inequalities and extending the Heinz norm inequalities to
more general cases with the help of a Bernstein type inequality for nonselfadjoint
operators, the convexity of norm functions, the Jensen functional and its properties, the
Hermite-Hadamard inequality, and so on. With this kind of research, the study of various
means inequalities, such as the geometric mean, the arithmetic mean, the Heinz mean,
arithmetic-geometric means, and Arithmetic-Geometric-Harmonic (A-G-H) weighted
means, has received much attention and development too. For recent interesting work
in this area, we refer the reader to [–] and references therein.
Based on [–], in this paper, we are concerned with the further refinements of the
geometric mean and the Heinz mean for operators in Hilbert spaces. Our purpose is to derive
some new generalizations of Heinz operator inequalities by refining the ordering relations
among Heinz means with different parameters, and of the geometric mean by
investigating geometric means of several operator variables in a weighted setting. Moreover, we will
obtain a matrix version of the Heinz inequality for the Hilbert-Schmidt norm.
Throughout this paper, B++(H) stands for the set of all bounded positive invertible
operators on a Hilbert space H, B(H) is the set of all bounded linear operators on H, and
B(H)sa is a convex domain of selfadjoint operators in B(H). For any T , S ∈ B++(H) and
ν ∈ [, ], we write
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T ν S := T T – ST – ν T
H(a, b) = H(a, b) = a + b ,
Hν (a, b) = H–ν (a, b), ν ∈ [, ],
√ab ≤ Hν (a, b) ≤ a + b , ν ∈ [, ].
H/(a, b) =
is called the Heinz operator mean of T and S. Clearly,
that is, the Heinz operator mean interpolates between the geometric mean and the
arithmetic mean.
2 Improved Heinz means inequalities
In a very recent work [], we establish the following inequalities:
Ht(T , S) ≤ – (( –– ts))
hold for s, t ∈ [, ] satisfying |s – | ≥ |t – |, s = .
In this section, we improve the result and give two theorems as follows.
≤ – ( – t) T S + ( – t)T ∇S.
Proof Writing
In view of the Taylor series of cosh x, we deduce that
≥ ,
With x–y instead of x, we have
+ ( – t) cosh ( – s) x – y
( – s)
Let a = ex, b = ey. Then
– (( –– ts)) x/ + (( –– ts)) x–s+ xs
With the positive operator T – ST – instead of x, we have
The proof is completed.
Fν (x) = ⎨⎧ xνl–ogxx–ν , x > , x = ,
x = ,
we have the following result.
It follows from the Taylor series of sinh x that
= +
β β β αx
≤ – α + α + α
+ ( – t) sinh(( – s)( x–y ))
( – s) ( – s)( x–y )
Put a = ex, b = ey. Then we get
a–tbt – atb–t
( – t)(log a – log b)
With the positive operator T – ST – instead of x, we have
≤ – (( –– ts))
T – ST – + ( – t)
( – s) s – Fs T – ST – .
≤ – (( –– ts))
3 Heinz inequality for the Hilbert-Schmidt norm
In this section, we let Mn be the Hilbert space of n × n complex matrices and let ·
stand for any unitarily invariant norm on Mn, i.e. UTV = T for all T ∈ Mn and for all
unitary matrices U, V ∈ Mn. We suppose that T , S, X ∈ Mn with T and S being positive
semidefinite. For T = [aij] ∈ Mn, the Hilbert-Schmidt norm of T is defined by
T =
Theorem . Let s, t ∈ [, ] satisfy
T tXS–t + T –tXSt
T /XS/ + ( – t) T sXS–s + T –sXSs
( – s)
.
Proof Noting that T and S are positive semidefinite, we know by the spectral theorem that
there exist unitary matrices U, V ∈ Mn such that
T = U U∗ and S = V V ∗,
Y = U∗XV = [yij].
Then we have
T tXS–t + T –tXSt
= U
(U U∗)tX(V V ∗)–t + (U U∗)–tX(V V ∗)t
(U tU∗)X(V –tV ∗) + (U –tU∗)X(V tV ∗)
U t(U∗XV ) –tV ∗ + U –t(U∗XV ) tV ∗
By a similar argument to the above, we deduce that
T /XS/ + ( – t) T sXS–s + T –sXSs
( – s)
( – s)
– ( – t)
By virtue of the inequalities (.) and (.), we obtain
– ( – t)
Thus, the proof is completed.
4 The inductive weighted geometric means and means inequalities
Let F : D → B(H) be a mapping of k variables defined in a convex domain D ⊆ B(H)k .
Recall from Hansen [] that F is regular if:
(i) The domain D is invariant under unitary transformations of H and
F U∗TU, . . . , U∗TkU = U∗F(T, . . . , Tk)U
for every (T, . . . , Tk) ∈ D and every unitary U on H.
(ii) Let P and Q be mutually orthogonal projections acting on H and take arbitrary
k-tuples (T, . . . , Tk) and (S, . . . , Sk) of operators in B(H) such that the compressed tuples
(PTP, . . . , PTkP) and (QSQ, . . . , QSkQ) are in the domain D. Then the k-tuple of diagonal
block matrices
(PTP + QSQ, . . . , PTkP + QSkQ)
is also in the domain D and
F(PTP + QSQ, . . . , PTkP + QSkQ)
Recall also from Hansen [] that the perspective of a regular operator mapping of several
variables is defined as
D+k = (T, . . . , Tk)|T, . . . , Tk > .
Now we prove another two properties of PF .
Theorem . Suppose that F : D+k → B(H)sa is regular, concave, and continuous. Then
the perspective function PF is monotone.
Proof Let Ti and Si be positive invertible operators such that Ti ≤ Si for i = , . . . , k + . If
Si – Ti is invertible for each i = , . . . , k + and λ ∈ (, ), then we have
λSi = λTi + ( – λ)Wi, i = , . . . , k + ,
Wi = λ( – λ)–(Si – Ti), i = , . . . , k + ,
are positive and invertible. Thus, the concavity of PF implies that
PF (S, . . . , Sk+) ≥ PF (T, . . . , Tk+).
νTi < Ti ≤ Si, i = , . . . , k + .
Then we have
PF (μT, . . . , μTk+) ≤ PF (S, . . . , Sk+).
Theorem . Suppose that F : D+k → B(H)sa is a regular, concave, and positively
homogeneous. Then the perspective function PF satisfies the property of congruence invariance:
for any invertible operator W on H.
Proof It follows from Theorem . of [] that the perspective function PF is concave.
Moreover, since F is positively homogeneous, it is easy to prove that PF is also positively
homogeneous. Hence, by Proposition . in [], we get the conclusion.
ik= βi = , and let αk+ ∈ [, ] and
Then, simulated by the significant work of Hansen [], we construct a sequence of
weighted multivariate geometric means Gα, Gα, . . . as follows.
(i) Let Gα(T ) = T for each positive definite invertible operator T .
(ii) To each weighted geometric mean Gkβ of k variables we associate an auxiliary
mapping Ak : D+k → B(H) such that Ak is regular and concave, and
Ak(T, . . . , Tk) = Gkβ (T, . . . , Tk)(–αk+) = T · · · Tkβk (–αk+)
β
for positive T, . . . , Tk , where β is the weight associated to T, . . . , Tk .
(iii) Define the geometric mean Gkα+ : D+k+ → B(H) of k + variables as
Gkα+(T, . . . , Tk+) = PAk (T, . . . , Tk+),
Particularly, the geometric means of two variables
coincide with the weighted geometric means of two variables T α T in the sense of Kubo
and Ando [], where α = (α, α) satisfy α + α = .
In the above procedure, αi is determined by βi and αk+ in the following sense:
and hence trace back to the case of k = . Therefore for fixed weight we can define the
corresponding weighted geometric mean.
α
Theorem . The means Gk : D+k → B(H)+ constructed as above are regular, positively
homogeneous, concave, and they satisfy
Gkα+(T, . . . , Tk, ) = Gkβ (T)(–αk+)
for T = (T, . . . Tk) ∈ D+.
k
Proof By the definition of Gkα , we know that Gkα for each k = , , . . . is the perspective of
a regular positively homogeneous map. Therefore, Gkα are regular and positively
homogeneous. Moreover, since Gkα+ is the perspective of (Gkβ )–αk+ , we see that (.) holds.
Next, we prove that Gkα is concave. Clearly, Gα is concave. Assume that Gkβ is concave for
some k and the corresponding weight β. For αk+ ∈ [, ], the map x → x–αk+ is operator
monotone (increasing) and operator concave. Then we have
Ak(T, . . . , Tk) = Gkβ (T, . . . , Tk)–αk+
is concave. Then by Theorem . in [] we see that its perspective Gkα+ is also concave. By
induction, we know that Gkα is concave for all k = , , . . . .
Remark . A similar analysis to Theorem . in [] shows that the above conditions
uniquely determine the Geometric means Gkα for k = , , . . . by setting Gα(T ) = T .
Theorem . Set T = (T, . . . , Tk) ∈ D+k. The means Gkα constructed as above have the
following properties:
(P) (consistency with scalars) Gkα(T) = T · · · Tkαk if the Ti’s commute;
α
α
(P) (joint homogeneity) Gkα(tT, . . . , tkTk) = tα · · · tk k Gα(T) for ti > ;
k
(P) (monotonicity) if Bi ≤ Ti for all ≤ i ≤ k, then Gα(B) ≤ Gα(T);
k k
(P) (congruence invariance) Gkα(W ∗TW , . . . , W ∗TkW ) = W ∗Gkα(T)W for any
invertible operator W on H;
(P) (self-duality) Gα(T–) = Gkα(T)–;
k
(P) (A-G-H weighted mean inequalities) ( ik= αiTi–)– ≤ Gkα(T) ≤ ik= αiTi;
(P) (determinant identity) det Gkα(T) = ik=(det Ti)αi .
Proof If T and T commute, then
Gα(T, T) = T T– TT– α T = Tα Tα .
Gkα+(T, . . . , Tk+) = PFk (T, . . . , Tk+)
= Tk+Fk Tk–+TTk–+, . . . , Tk–+TkTk+ Tk+
–
= Tk+Gkβ Tk–+TTk–+, . . . , Tk–+TkTk+
– (–αk+)Tk+
= Tk+
= Tβ(–αk+) · · · Tkβk(–αk+)Tk+–(–αk+)
we see that (P) also holds for k + . By induction, we know that (P) holds for k = , , . . . .
It is easy to verify (P) holds for k = and k = . Assume that (P) holds for some k > .
Then we have
By induction, we get
= tk+ tk–+tβ · · · tkβk (–αk+)Gkα+(T, . . . , Tk , Tk+)
= tα · · · tk k tkα+k+ Gkα+(T, . . . , Tk , Tk+).
α
Hence (P) is true.
(P) and (P) follow from Theorems . and ..
Clearly, (P) is true for k = and k = . Assume that (P) is true for some k > . Then we
have
α
Gk+ T–, . . . , Tk–, Tk–+
= Tk–+Fk Tk+T–Tk+, . . . , Tk+Tk–Tk+ Tk–+
= Tk–+Gkβ Tk+T–Tk+, . . . , Tk+Tk–Tk+ (–αk+)Tk–+.
By induction, we get
= Tk–+Gkβ Tk–+TTk–+, . . . , Tk–+Tk Tk+
– –(–αk+)Tk–+
= Tk+Gkβ Tk–+TTk–+, . . . , Tk–+Tk Tk–+ (–αk+)Tk+ –
= Gkα+(T, . . . , Tk , Tk+)–,
which verifies (P).
The A-G-H weighted mean inequality, i.e. the arithmetic-geometric-harmonic weighted
mean inequality reads
Xp ≤ + p(X – )
for arbitrary (T, . . . , Tk ) ∈ D+k. Firstly, we show the second inequality. It is easy to see the
second inequality holds for k = . Assume the inequality holds for some k. Then, by virtue
of
Now taking perspective, we have
α
Gk+(T, . . . , Tk , Tk+) = PFk (T, . . . , Tk , Tk+)
= Tk+Fk Tk–+TTk–+, . . . , Tk–+Tk Tk+ Tk+
–
≤ Tk+
αiTk–+TiTk–+ + αk+ Tk+
we infer that
which means that (P) is true.
By induction, the second inequality is proved. Next, it follows from the second inequality
By inversion and using the self-duality (P) of the weighted geometric mean, we get
≤ Gk
Hence the property (P) holds.
α β
Gk+(T, . . . , Tk , Tk+) = Tk+ k
G
Tk–+TTk–+, . . . , Tk–+Tk Tk+
– (–αk+)Tk+,
1. Abbas , H, Mourad , B: A family of refinements of Heinz inequalities of matrices . J. Inequal. Appl . 2014 , 267 ( 2014 )
2. Fujii , JI, Fujii , M, Seo, Y, Zuo , HL : Recent developments of matrix versions of the arithmetic-geometric mean inequality . Ann. Funct. Anal . 7 ( 1 ), 102 - 117 ( 2016 )
3. Malekinejad , S, Talebi, S, Ghazanfari, AG : Reverses of Young and Heinz inequalities for positive linear operators . J. Inequal. Appl . 2016 , 35 ( 2016 )
4. Manjegani , SM: Tracial and majorisation Heinz mean-type inequalities for matrices . J. Inequal. Appl . 2016 , 23 ( 2016 )
5. Krnic´ , M, Pecˇaric´, J: Improved Heinz inequalities via the Jensen functional. Open Math . 11 , 1698 - 1710 ( 2013 )
6. Kubo , F, Ando, T: Means of positive linear operators . Math. Ann. 246 , 205 - 224 ( 1980 )
7. Hansen , F: Regular operator mappings and multivariate geometric means . Linear Algebra Appl . 461 , 123 - 138 ( 2014 )
8. Liang , J, Shi, G : Refinements of the Heinz operator inequalities . Linear Multilinear Algebra 63 , 1337 - 1344 ( 2015 )
9. Zuo , HL, Cheng , N : Improved reverse arithmetic-geometric means inequalities for positive operators on Hilbert space . Math. Inequal. Appl . 18 ( 1 ), 51 - 60 ( 2015 )
10. Zuo , HL, Duan, GC : Some inequalities of operator monotone functions . J. Math. Inequal . 8 ( 4 ), 777 - 781 ( 2014 )