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

Abstract

Abstract 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 n × n $n\times n$ matrices, then ∥ A X B ∗ ∥ 2 ≤ ∥ f 1 ( A ∗ A ) X g 1 ( B ∗ B ) ∥ ∥ f 2 ( A ∗ A ) X g 2 ( B ∗ B ) ∥ , $$\begin{aligned} \bigl\Vert AXB^{*} \bigr\Vert ^{2}\leq \bigl\Vert f_{1} \bigl(A^{*}A\bigr)Xg_{1}\bigl(B^{*}B\bigr) \bigr\Vert \bigl\Vert f_{2}\bigl(A^{*}A\bigr)Xg_{2}\bigl(B^{*}B\bigr) \bigr\Vert , \end{aligned}$$ where f 1 $f_{1}$ , f 2 $f_{2}$ , g 1 $g_{1}$ , g 2 $g_{2}$ are non-negative continuous functions such that f 1 ( t ) f 2 ( t ) = t $f_{1}(t)f_{2}(t)=t$ and g 1 ( t ) g 2 ( t ) = t $g_{1}(t)g_{2}(t)=t$ ( t ≥ 0 $t\geq0$ ). We also obtain the inequality | | | A B ∗ | | | 2 ≤ | | | p ( A ∗ A ) m p + ( 1 − p ) ( B ∗ B ) s 1 − p | | | | | | ( 1 − p ) ( A ∗ A ) n 1 − p + p ( B ∗ B ) t p | | | , $$\begin{aligned} \bigl\vert \!\bigl\vert \!\bigl\vert AB^{*} \bigr\vert \!\bigr\vert \!\bigr\vert ^{2} &\leq \bigl\vert \!\bigl\vert \!\bigl\vert p \bigl(A^{*}A\bigr)^{\frac{m}{p}}+ (1-p) \bigl(B^{*}B\bigr)^{\frac {s}{1-p}} \bigr\vert \!\bigr\vert \!\bigr\vert \bigl\vert \!\bigl\vert \!\bigl\vert (1-p) \bigl(A^{*}A\bigr)^{\frac{n}{1-p}}+ p\bigl(B^{*}B \bigr)^{\frac{t}{p}} \bigr\vert \!\bigr\vert \!\bigr\vert , \end{aligned}$$ in which m, n, s, t are real numbers such that m + n = s + t = 1 $m+n=s+t=1$ , | | | ⋅ | | | $\vert \!\vert \!\vert \cdot \vert \!\vert \!\vert $ is an arbitrary unitarily invariant norm and p ∈ [ 0 , 1 ] $p\in[0,1]$ .

Keywords

• arithmetic-geometric mean
• unitarily invariant norm
• Hilbert-Schmidt norm
• Cauchy-Schwarz inequality