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Some means inequalities for positive operators in Hilbert spaces
Journal of Inequalities and Applications volume 2017, Article number: 14 (2017)
Abstract
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.
1 Introduction
Since Heinz proved a series of useful norm inequalities, which are closely related to the Cordes inequality and the Furuta inequality, in 1951, 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 [1–10] and references therein.
Based on [1–8], 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, \(\mathfrak{B}^{++}(\mathcal{H})\) stands for the set of all bounded positive invertible operators on a Hilbert space \(\mathcal{H}\), \(\mathfrak{B}(\mathcal{H})\) is the set of all bounded linear operators on \(\mathcal{H}\), and \(\mathfrak{B}(\mathcal{H})_{\mathrm{sa}}\) is a convex domain of selfadjoint operators in \(\mathfrak{B}(\mathcal {H})\). For any \(T,S\in\mathfrak{B}^{++}(\mathcal{H}) \) and \(\nu\in [0,1]\), we write
and
as the geometric mean of T and S. When \(\nu=1/2\) we write \(T\nabla S\) and \(T\sharp S\) in short, respectively. We refer the reader to Kubo and Ando [6] for more information on the means of positive linear operators.
Recall that, for any \(a,b\geq0\), the number
is called the Heinz mean \(H_{\nu}(a,b)\) of a and b. It is clear that
For any \(T,S\in\mathfrak{B}^{++}(\mathcal{H})\) and \(\nu\in[0,1]\), the operator
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 [8], we establish the following inequalities:
and
hold for \(s,t\in[0,1]\) satisfying \(\vert s-\frac{1}{2}\vert \geq \vert t-\frac {1}{2}\vert \), \(s\neq\frac{1}{2}\).
In this section, we improve the result and give two theorems as follows.
Theorem 2.1
Suppose \(T,S\in\mathfrak{B}^{++}(\mathcal{H})\), and let \(s,t\in[0,1]\) satisfy
Then
Proof
Writing
we see
In view of the Taylor series of coshx, we deduce that
i.e.,
With \(\frac{x-y}{2}\) instead of x, we have
Let \(a=e^{x}\), \(b=e^{y}\). Then
Taking \(a=x\) and \(b=1\) in the inequality (2.2), we get
With the positive operator \(T^{-\frac{1}{2}}ST^{-\frac{1}{2}}\) instead of x, we have
The proof is completed. □
For the functions \(F_{\nu}:\mathbb{R}_{+}\rightarrow\mathbb{R}\) \((\nu\in [0,1])\) defined by
we have the following result.
Theorem 2.2
Suppose \(T,S\in\mathfrak{B}^{++}(\mathcal{H})\) and let \(s,t\in[0,1]\) satisfy
Then
Proof
Writing \(\alpha=1-2s\), \(\beta=1-2t\), we have
It follows from the Taylor series of sinhx that
With \(\frac{x-y}{2}\) instead of x, we know that, for any \(s,t\in [0,1]\) satisfying \(\vert s-\frac{1}{2}\vert \geq \vert t-\frac{1}{2}\vert \), \(s, t \neq\frac{1}{2}\),
Put \(a=e^{x}\), \(b=e^{y}\). Then we get
Letting \(a=x\) and \(b=1\) in inequality (2.3), we see that
that is,
With the positive operator \(T^{-\frac{1}{2}}ST^{-\frac{1}{2}}\) instead of x, we have
Therefore,
□
3 Heinz inequality for the Hilbert-Schmidt norm
In this section, we let \(M_{n}\) be the Hilbert space of \(n\times n\) complex matrices and let \(\Vert \cdot \Vert \) stand for any unitarily invariant norm on \(M_{n}\), i.e. \(\Vert UTV\Vert =\Vert T\Vert \) for all \(T\in M_{n}\) and for all unitary matrices \(U,V\in M_{n}\). We suppose that \(T, S, X \in M_{n}\) with T and S being positive semidefinite. For \(T=[a_{ij}]\in M_{n}\), the Hilbert-Schmidt norm of T is defined by
It is well known that the Hilbert-Schmidt norm is unitarily invariant.
Next, we prove the following matrix version of Heinz inequality for the Hilbert-Schmidt norm.
Theorem 3.1
Let \(s,t\in[0,1]\) satisfy
Then
Proof
Noting that T and S are positive semidefinite, we know by the spectral theorem that there exist unitary matrices \(U, V \in M_{n}\) such that
where
Put
Then we have
Hence,
By a similar argument to the above, we deduce that
and
By virtue of the inequalities (2.1) and (2.2), we obtain
Thus, the proof is completed. □
4 The inductive weighted geometric means and means inequalities
Let \(F: \mathcal{D}\rightarrow\mathfrak{B}(\mathcal{H})\) be a mapping of k variables defined in a convex domain \(\mathcal{D}\subseteq \mathfrak{B}(\mathcal{H})^{k}\). Recall from Hansen [7] that F is regular if:
(i) The domain \(\mathcal{D}\) is invariant under unitary transformations of \(\mathcal{H}\) and
for every \((T_{1},\ldots, T_{k})\in\mathcal{D}\) and every unitary U on \(\mathcal{H}\).
(ii) Let P and Q be mutually orthogonal projections acting on \(\mathcal{H}\) and take arbitrary k-tuples \((T_{1},\ldots ,T_{k})\) and \((S_{1},\ldots,S_{k})\) of operators in \(\mathfrak{B}(\mathcal{H})\) such that the compressed tuples \((PT_{1}P,\ldots,PT_{k}P)\) and \((QS_{1}Q, \ldots, QS_{k}Q)\) are in the domain \(\mathcal{D}\). Then the k-tuple of diagonal block matrices
is also in the domain \(\mathcal{D}\) and
Recall also from Hansen [7] that the perspective of a regular operator mapping of several variables is defined as
Hansen [7] proves that the perspective \(\mathcal{P}_{F}\) of a convex regular map \(F:\mathcal{D}^{k}_{+}\rightarrow\mathfrak{B}(\mathcal {H})\) is regular and convex.
Write
Now we prove another two properties of \(\mathcal{P}_{F}\).
Theorem 4.1
Suppose that \(F: \mathcal{D}^{k}_{+}\rightarrow\mathfrak {B}(\mathcal{H})_{\mathrm{sa}}\) is regular, concave, and continuous. Then the perspective function \(\mathcal{P}_{F}\) is monotone.
Proof
Let \(T_{i}\) and \(S_{i}\) be positive invertible operators such that \(T_{i}\leq S_{i}\) for \(i=1,\ldots, k+1\). If \(S_{i}-T_{i}\) is invertible for each \(i=1,\ldots, k+1\) and \(\lambda\in(0,1)\), then we have
where
are positive and invertible. Thus, the concavity of \(\mathcal{P}_{F}\) implies that
For \(\lambda\rightarrow1\), by continuity, we get
Generally, choose \(0<\nu<1\) such that
Then we have
Letting \(\nu\rightarrow1\), we get the conclusion. □
Theorem 4.2
Suppose that \(F: \mathcal{D}^{k}_{+}\rightarrow\mathfrak {B}(\mathcal{H})_{\mathrm{sa}}\) is a regular, concave, and positively homogeneous. Then the perspective function \(\mathcal{P}_{F}\) satisfies the property of congruence invariance:
for any invertible operator W on \(\mathcal{H}\).
Proof
It follows from Theorem 3.2 of [7] that the perspective function \(\mathcal{P}_{F}\) is concave. Moreover, since F is positively homogeneous, it is easy to prove that \(\mathcal{P}_{F}\) is also positively homogeneous. Hence, by Proposition 2.3 in [7], we get the conclusion. □
Let
with \(\sum_{i=1}^{k}\beta_{i}=1\), and let \(\alpha_{k+1}\in[0,1]\) and
Then, simulated by the significant work of Hansen [7], we construct a sequence of weighted multivariate geometric means \(G^{\alpha }_{1}, G^{\alpha}_{2},\ldots\) as follows.
(i) Let \(G^{\alpha}_{1}(T)=T\) for each positive definite invertible operator T.
(ii) To each weighted geometric mean \(G_{k}^{\beta}\) of k variables we associate an auxiliary mapping \(A_{k}: \mathcal {D}^{k}_{+}\rightarrow\mathfrak{B}(\mathcal{H})\) such that \(A_{k}\) is regular and concave, and
for positive \(T_{1},\ldots, T_{k}\), where β is the weight associated to \(T_{1},\ldots, T_{k}\).
(iii) Define the geometric mean \(G_{k+1}^{\alpha}: \mathcal {D}^{k+1}_{+}\rightarrow\mathfrak{B}(\mathcal{H})\) of \(k+1\) variables as
where
Particularly, the geometric means of two variables
coincide with the weighted geometric means of two variables \(T_{1}\sharp _{\alpha_{1}}T_{2}\) in the sense of Kubo and Ando [6], where \(\alpha=(\alpha_{1},\alpha_{2})\) satisfy \(\alpha_{1}+\alpha_{2}=1\).
In the above procedure, \(\alpha_{i} \) is determined by \(\beta_{i} \) and \(\alpha_{k+1}\) in the following sense:
Conversely for fixed \(\alpha=(\alpha_{1},\ldots, \alpha_{k+1})\), we can set
and hence trace back to the case of \(k=1\). Therefore for fixed weight we can define the corresponding weighted geometric mean.
Theorem 4.3
The means \(G_{k}^{\alpha}: \mathcal{D}^{k}_{+}\rightarrow \mathfrak{B}(\mathcal{H})^{+}\) constructed as above are regular, positively homogeneous, concave, and they satisfy
for \(\mathbb{T}=(T_{1},\ldots T_{k})\in \mathcal{D}^{k}_{+}\).
Proof
By the definition of \(G_{k}^{\alpha}\), we know that \(G_{k}^{\alpha}\) for each \(k=2,3,\ldots\) is the perspective of a regular positively homogeneous map. Therefore, \(G_{k}^{\alpha}\) are regular and positively homogeneous. Moreover, since \(G_{k+1}^{\alpha}\) is the perspective of \((G_{k}^{\beta})^{1-\alpha_{k+1}}\), we see that (4.2) holds.
Next, we prove that \(G_{k}^{\alpha}\) is concave. Clearly, \(G_{1}^{\alpha}\) is concave. Assume that \(G^{\beta}_{k}\) is concave for some k and the corresponding weight β. For \(\alpha_{k+1}\in[0,1]\), the map \(x\rightarrow x^{1-\alpha_{k+1}}\) is operator monotone (increasing) and operator concave. Then we have
where \(\mathbb{T}=(T_{1},\ldots, T_{k})\), \(\mathbb{S}=(S_{1},\ldots, S_{k})\). So the auxiliary mapping
is concave. Then by Theorem 3.2 in [7] we see that its perspective \(G_{k+1}^{\alpha}\) is also concave. By induction, we know that \(G^{\alpha}_{k}\) is concave for all \(k=1,2,\ldots\) . □
Remark 4.4
A similar analysis to Theorem 4.3 in [7] shows that the above conditions uniquely determine the Geometric means \(G_{k}^{\alpha}\) for \(k=1,2,\ldots\) by setting \(G_{1}^{\alpha}(T)=T\).
Theorem 4.5
Set \(\mathbb{T}=(T_{1},\ldots, T_{k})\in\mathcal{D}^{k}_{+}\). The means \(G_{k}^{\alpha}\) constructed as above have the following properties:
-
(P1)
(consistency with scalars) \(G_{k}^{\alpha}(\mathbb {T})=T_{1}^{\alpha_{1}}\cdots T_{k}^{\alpha_{k}}\) if the \(T_{i}\) ’s commute;
-
(P2)
(joint homogeneity) \(G_{k}^{\alpha}(t_{1} T_{1},\ldots ,t_{k} T_{k}) =t_{1}^{\alpha_{1}}\cdots t_{k}^{\alpha_{k}}G_{k}^{\alpha}(\mathbb {T})\) for \(t_{i}>0 \);
-
(P3)
(monotonicity) if \(B_{i}\leq T_{i}\) for all \(1\leq i\leq k\), then \(G_{k}^{\alpha}(\mathbb{B}) \leq G_{k}^{\alpha}(\mathbb{T})\);
-
(P4)
(congruence invariance) \(G_{k}^{\alpha}(W^{*} T_{1} W,\ldots, W^{*} T_{k} W)=W^{*}G_{k}^{\alpha}(\mathbb{T}) W\) for any invertible operator W on \(\mathcal{H}\);
-
(P5)
(self-duality) \(G_{k}^{\alpha}(\mathbb {T}^{-1})=G_{k}^{\alpha}(\mathbb{T})^{-1}\);
-
(P6)
(A-G-H weighted mean inequalities) \((\sum_{i=1}^{k} \alpha_{i} T_{i}^{-1})^{-1}\leq G_{k}^{\alpha}(\mathbb{T})\leq\sum_{i=1}^{k} \alpha_{i} T_{i}\);
-
(P7)
(determinant identity) \(\det G_{k}^{\alpha }(\mathbb{T})=\Pi_{i=1}^{k}(\det T_{i})^{\alpha_{i}}\).
Proof
If \(T_{1}\) and \(T_{2}\) commute, then
Hence, (P1) holds for \(k=1,2\). Now assume that (P1) holds for some \(k>2\). Since
we see that (P1) also holds for \(k+1\). By induction, we know that (P1) holds for \(k=1,2,\ldots\) .
It is easy to verify (P2) holds for \(k=1\) and \(k=2\). Assume that (P2) holds for some \(k>2\). Then we have
By induction, we get
Hence (P2) is true.
(P3) and (P4) follow from Theorems 4.1 and 4.2.
Clearly, (P5) is true for \(k=1\) and \(k=2\). Assume that (P5) is true for some \(k>2\). Then we have
By induction, we get
which verifies (P5).
The A-G-H weighted mean inequality, i.e. the arithmetic-geometric-harmonic weighted mean inequality reads
for arbitrary \((T_{1},\ldots,T_{k})\in\mathcal{D}^{k}_{+}\). Firstly, we show the second inequality. It is easy to see the second inequality holds for \(k=1\). Assume the inequality holds for some k. Then, by virtue of
for \(X\in\mathcal{D}^{1}\) (the set of positive operators) and \(p\in [0,1]\), we obtain
Now taking perspective, we have
By induction, the second inequality is proved. Next, it follows from the second inequality that
By inversion and using the self-duality (P7) of the weighted geometric mean, we get
Hence the property (P6) holds.
For \(T\in\mathcal{D}^{1}\) and \(p\in\mathbb{R}\), we have \(\det T^{p}=(\det T)^{p}\) due to \(\det T=\exp(\textrm{Tr}\log T)\). For \(k=1\) and \(k=2\), (P7) is obviously correct. Assume that (P7) holds for some \(k>2\). Then, using
we infer that
which means that (P7) is true. □
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Acknowledgements
The authors acknowledge support from NSFC (No. 11571229). The authors also would like to thank the reviewers very much for their useful suggestions.
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Liang, J., Shi, G. Some means inequalities for positive operators in Hilbert spaces. J Inequal Appl 2017, 14 (2017). https://doi.org/10.1186/s13660-016-1283-x
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DOI: https://doi.org/10.1186/s13660-016-1283-x