Improved arithmetic-geometric and Heinz means inequalities for Hilbert space operators. (English) Zbl 1275.47038

Kittaneh, Fuad; Krnić, Mario; Lovričević, Neda; Pečarić, Josip

Publ. Math. 80, No. 3-4, 465-478 (2012).

For positive invertible operators \(A\), \(B\) on a Hilbert space \(\mathcal {H}\), the weighted arithmetic-geometric mean inequality holds: \(A\#_\alpha B\leq A\nabla _\alpha B\), \(\alpha \in [0,1]\). Here, \(A\nabla _\alpha B=(1-\alpha)A+\alpha B\) and \(A\#_\alpha B=A^{1/2}(A^{-1/2}BA^{-1/2})^{\alpha }A^{1/2}\). \(A\nabla _\alpha B\) is called the weighted arithmetic mean of \(A,B\) and \(A\#_\alpha B\) is called the weighted geometric mean of \(A, B\). Let \(B\) be a positive invertible operator on \(\mathcal{H}\), with \(C\) an invertible operator on \(\mathcal{H}\), \(A=C^*C\) and \((p_1,p_2) \in \mathbb{R}_+^2\). In this paper, the authors prove the inequalities: \[ 2\max \{p_1,p_2\}[A\nabla _{1/2} B-C^*(C^{*-1}BC^{-1})^{1/2}C] \] \[ \geq (p_1+p_2)[A\nabla _{\frac{p_1}{p_1+p_2}}B-C^*( C^{*-1}BC^{-1})^{\frac{p_1}{p_1+p_2}}C] \] \[ \geq 2\min \{p_1,p_2\}[A\nabla _{1/2} B-C^*(C^{*-1}BC^{-1})^{1/2}C]. \] This generalizes a similar inequality proved by F. Kittaneh and Y. Manasrah [Linear Multilinear Algebra 59, No. 9, 1031–1037 (2011; Zbl 1225.15022)]. Several related inequalities are discussed. As an application, it is shown that the Heinz mean interpolates between the arithmetic mean and the geometric mean.

Reviewer: Jaspal Singh Aujla (Jalandhar)

MSC:

47A63	Linear operator inequalities
26D15	Inequalities for sums, series and integrals
47A64	Operator means involving linear operators, shorted linear operators, etc.

Keywords:

Hilbert space; positive invertible operator; arithmetic operator mean; geometric operator mean; Heinz operator mean; superadditivity; monotonicity; refinement; converse