Extensions of interpolation between the arithmetic-geometric mean inequality for matrices

In this paper, we present some extensions of interpolation between the arithmetic-geometric means inequality. Among other inequalities, it is shown that if A, B, X are \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n\times n$\end{document}n×n matrices, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \bigl\Vert AXB^{*} \bigr\Vert ^{2}\leq \bigl\Vert f_{1} \bigl(A^{*}A\bigr)Xg_{1}\bigl(B^{*}B\bigr) \bigr\Vert \bigl\Vert f_{2}\bigl(A^{*}A\bigr)Xg_{2}\bigl(B^{*}B\bigr) \bigr\Vert , \end{aligned}$$ \end{document}∥AXB∗∥2≤∥f1(A∗A)Xg1(B∗B)∥∥f2(A∗A)Xg2(B∗B)∥, where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f_{1}$\end{document}f1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f_{2}$\end{document}f2, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g_{1}$\end{document}g1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g_{2}$\end{document}g2 are non-negative continuous functions such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$f_{1}(t)f_{2}(t)=t$\end{document}f1(t)f2(t)=t and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g_{1}(t)g_{2}(t)=t$\end{document}g1(t)g2(t)=t (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t\geq0$\end{document}t≥0). We also obtain the inequality \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{aligned} \bigl\vert \!\bigl\vert \!\bigl\vert AB^{*} \bigr\vert \!\bigr\vert \!\bigr\vert ^{2} &\leq \bigl\vert \!\bigl\vert \!\bigl\vert p \bigl(A^{*}A\bigr)^{\frac{m}{p}}+ (1-p) \bigl(B^{*}B\bigr)^{\frac {s}{1-p}} \bigr\vert \!\bigr\vert \!\bigr\vert \bigl\vert \!\bigl\vert \!\bigl\vert (1-p) \bigl(A^{*}A\bigr)^{\frac{n}{1-p}}+ p\bigl(B^{*}B \bigr)^{\frac{t}{p}} \bigr\vert \!\bigr\vert \!\bigr\vert , \end{aligned}$$ \end{document}|||AB∗|||2≤|||p(A∗A)mp+(1−p)(B∗B)s1−p||||||(1−p)(A∗A)n1−p+p(B∗B)tp|||, in which m, n, s, t are real numbers such that \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$m+n=s+t=1$\end{document}m+n=s+t=1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\vert \!\vert \!\vert \cdot \vert \!\vert \!\vert $\end{document}|||⋅||| is an arbitrary unitarily invariant norm and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$p\in[0,1]$\end{document}p∈[0,1].


Introduction and preliminaries
Let M n be the C * -algebra of all n×n complex matrices and • , • be the standard scalar product in C n with the identity I.The Gelfand map f (t) → f (A) is an isometrical *isomorphism between the C * -algebra C(sp(A)) of continuous functions on the spectrum sp(A) of a Hermitian matrix A and the C * -algebra generated by A and I.
A norm ||| • ||| on M n is said to be unitarily invariant norm if |||UAV ||| = |||A|||, for all unitary matrices U and V .For A ∈ M n , let s denote the singular values of A, i.e. the eigenvalues of the positive semidefinite matrix |A| = (A * A) 1 2 arranged in a decreasing order with their multiplicities counted.Note that s j (A) = s j (A * ) = s j (|A|) (1 ≤ j ≤ n) and A = s 1 (A).The Ky Fan norm of a matrix A is defined as The Fan dominance theorem asserts that A (k) ≤ B (k) for k = 1, 2, • • • , n if and only if |||A||| ≤ |||B||| for every unitarily invariant norm (see [8, p.93]).The Hilbert-Schmidt norm is defined by , where is unitarily invariant.
The classical Cauchy-Schwarz inequality for [11].Bhatia and Davis gave a matrix Cauchy-Schwarz inequality as follows where A, B, X ∈ M n .(For further information about the Cauchy-Schwarz inequality, see [4,6,7] and references therein.)Recently, Kittaneh et al. [3] extended inequality (1.1) to the form where A, B, X ∈ M n and p ∈ [0, 1].Audenaert [2] proved that for all A, B ∈ M n and all p ∈ [0, 1], we have In [14], the authors generalized inequality (1.3) for all A, B, X ∈ M n and all p ∈ [0, 1] to the form Inequality (1.4) interpolates between the arithmetic-geometric mean inequality.In [3], the authors showed a refinement of inequality (1.4) for the Hilbert-Schmidt norm as follows in which A, B, X ∈ M n , p ∈ [0, 1] and r = min{p, 1 − p}.The Young inequality for every unitarily invariant norm states that are positive definite matrices and p ∈ [0, 1] (see [1] and also [5]).Kosaki [10] extended the last inequality for the Hilbert-Schmidt norm as follows where A, B are positive definite matrices, X is any matrix and p ∈ [0, 1].In [9], the authors considered as a refined matrix Young inequality for the Hilbert-Schmidt norm in which A, B are positive semidefinite matrices, X ∈ M n , p ∈ [0, 1] and r = min{p, 1− p}.
Among other results, we obtain a refinement of inequality (1.5) and we also extend inequalities (1.2), (1.3) and (1.5) for the function f (t) = t p (p ∈ R).

Main results
In this section, by using some ideas of [3] we extend the Audenaert results for the operator norm. (2.1) Proof.It follows from that we get the desired result.
Corollary 2.2.If A, B, X ∈ M n and m, n, s, t are real numbers such that m + n = In the next results, we show some generalizations of inequality (1.3) for the operator norm.
Proof.Using Theorem 2.1 for X = I we have (by the Young inequality).
Corollary 2.4.Let A, B ∈ M n and let f, g be non-negative continues functions such Proof.Applying Theorem 2.1 and the Young inequality we get

2
(by Theorem 2.1 for f and √ g) (by the Young inequality).

Some interpolations for unitarily invariant norms
In this section, by applying some ideas of [3] we generalize some interpolations for an arbitrary unitarily invariant norm.
Let Q k,n denote the set of all strictly increasing k-tuples chosen from 1, 2, • • • , n, i.e.
The following lemma gives some properties of the kth antisymmetric tensor powers of matrices in M n ; see [8, p.18].
Now, we show inequality (2.2) for an arbitrary unitarily invariant norm.
Theorem 3.2.Let A, B, X ∈ M n and ||| • ||| be an arbitrary unitarily invariant norm. Then where m, n, s, t are real numbers such that m + n = s + t = 1.In particular, if A, B are positive definite where p ∈ [0, 1].
Proof.If we replace A, B and X by ∧ k A, ∧ k B and ∧ k X, their kth antisymmetric tensor powers in inequality (2.1) and apply Lemma 3.1 , then we have that is equivalent to Applying Lemma 3.1(d), we have where (by the Cauchy-Schwarz inequality), where where A, B ∈ M n , m, n, s, t are real numbers such that m and These inequalities are generalizations of (1.4) for the Hilbert-Schmidt norms.
In the following theorem, we show a refinement of inequality (3.5) for the Hilbert-Schmidt norm.
Proof.Using inequality (3.1), we obtain where p ∈ [0, 1] and r = min{p, 1 − p}, and the proof is complete.Theorem 3.6 includes a special case as follows.
Proof.For p ∈ [0, 1], if we put m = t = p and n = s = 1 − p in Theorem 3.6, then we get the desired result.
The next result is a refinement of inequality (1.5).
Proof.The proof of inequality (3.7) is similar to that of inequality (3.6).Thus, we only need to prove the inequality (3.6).If 0 < p ≤ 1 2 , then we replace A and B by A * A and B * B in inequality (1.8), respectively, we have (3.8) Interchanging the roles of p and 1 − p in the inequality (3.8), we get (ii) for where r = min{p, 1 − p} and r 0 = min{2r, 1 − 2r}.
The following lemma has been shown in [12], and considered as a refined matrix Youngs inequality for every unitary invariant norm.where r 0 = min{p, 1 − p}.

Corollary 3 . 4 .
Let A, B ∈ M n , m, n, s, t be real numbers such that m + n = s + t = 2 and ||| • ||| be an arbitrary unitarily invariant norm.Then