Some means inequalities for positive operators in Hilbert spaces

Journal of Inequalities and Applications, Jan 2017

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. MSC: 47A30, 47A63, 26D10, 26D20, 26E60.

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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 © The Author(s) 2017. 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. 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–ogxx–ν , x > , x = , x = , we have the following result. It follows from the Taylor series of sinh x that =  + β β β αx ≤  – 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 tU∗)X(V –tV ∗) + (U –tU∗)X(V tV ∗)  U t(U∗XV ) –tV ∗ + U –t(U∗XV ) tV ∗  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∗TU, . . . , 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 (PTP, . . . , PTkP) and (QSQ, . . . , QSkQ) are in the domain D. Then the k-tuple of diagonal block matrices (PTP + QSQ, . . . , PTkP + QSkQ) is also in the domain D and F(PTP + QSQ, . . . , 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 Gα, Gα, . . . as follows. (i) Let Gα(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, Gα 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 Gα(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(tT, . . . , tkTk) = tα · · · 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 ∗TW , . . . , 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 Gα(T, T) = T T–  TT–  α T = Tα Tα .  Gkα+(T, . . . , Tk+) = PFk (T, . . . , Tk+) = Tk+Fk Tk–+TTk–+, . . . , Tk–+TkTk+ Tk+  –   = Tk+Gkβ Tk–+TTk–+, . . . , Tk–+TkTk+  –  (–αk+)Tk+  = Tk+ = Tβ(–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–+tβ · · · tkβk (–αk+)Gkα+(T, . . . , Tk , Tk+) = tα · · · 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–+TTk–+, . . . , Tk–+Tk Tk+ –  –(–αk+)Tk–+ = Tk+Gkβ Tk–+TTk–+, . . . , 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–+TTk–+, . . . , Tk–+Tk Tk+ Tk+  –   ≤ Tk+ αiTk–+TiTk–+ + αk+ Tk+  we infer that which means that (P) is true. 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Jin Liang, Guanghua Shi. Some means inequalities for positive operators in Hilbert spaces, Journal of Inequalities and Applications, 2017, 14, DOI: 10.1186/s13660-016-1283-x