Several new inequalities on operator means of non-negative maps and Khatri–Rao products of positive definite matrices

Let A_i > 0 and B_i > 0(i = 1,2) be n × n compatible partitioned matrices. Then for any real number α,

(i) In order to see if this indeed is true, let $D_1=A_1^{-1/2}{A}_2{A}_1^{-1/2}$ and $D_2=B_1^{-1/2}{B}_2{B}_1^{-1/2}$ . Then $$\begin{array}{c}(A_1\mathrm{\Theta }{B}_1)\underset{\alpha }{\#}(A_2\mathrm{\Theta }{B}_2)={(A_1\mathrm{\Theta }{B}_1)}^{1/2}{({(A_1\mathrm{\Theta }{B}_1)}^{-1/2}(A_2\mathrm{\Theta }{B}_2){(A_1\mathrm{\Theta }{B}_1)}^{-1/2})}^{\alpha }{(A_1\mathrm{\Theta }{B}_1)}^{1/2}\hfill \\ =\left(A_1^{1/2}\mathrm{\Theta }{B}_1^{1/2}\right){\left(\left(A_1^{-1/2}\mathrm{\Theta }{B}_1^{-1/2}\right)(A_2\mathrm{\Theta }{B}_2)\left(A_1^{-1/2}\mathrm{\Theta }{B}_1^{-1/2}\right)\right)}^{\alpha }\left(A_1^{1/2}\mathrm{\Theta }{B}_1^{1/2}\right)\hfill \\ =\left(A_1^{1/2}\mathrm{\Theta }{B}_1^{1/2}\right)\left\{{\left(A_1^{-1/2}{A}_2{A}_1^{-1/2}\right)}^{\alpha }\mathrm{\Theta }{\left(B_1^{-1/2}{B}_2{B}_1^{-1/2}\right)}^{\alpha }\right\}\left(A_1^{1/2}\mathrm{\Theta }{B}_1^{1/2}\right)\hfill \\ =\left(A_1^{1/2}\mathrm{\Theta }{B}_1^{1/2}\right)\left(D_1^{\alpha }\mathrm{\Theta }{D}_2^{\alpha }\right)\left(A_1^{1/2}\mathrm{\Theta }{B}_1^{1/2}\right)\hfill \\ =\left(A_1^{1/2}{D}_1^{\alpha }{A}_1^{1/2}\right)\mathrm{\Theta }\left(B_1^{1/2}{D}_2^{\alpha }{B}_1^{1/2}\right)\hfill \\ =(A_1\underset{\alpha }{\#}{A}_2)\mathrm{\Theta }(B_1\underset{\alpha }{\#}{B}_2)\text{.}\hfill \end{array}$$ Similarly, we can prove part (ii).

Let A_i > 0 (i = 1,2) be n × n compatible partitioned matrices. Then

Due to Lemma 2.1 and 1-27, we have $$\begin{array}{c}{(A_1\mathrm{\Theta }{A}_2)}^p\underset{\alpha }{\#}{(A_1\mathrm{\Theta }{A}_2)}^q=\left(A_1^p\mathrm{\Theta }{A}_2^p\right)\underset{\alpha }{\#}\left(A_1^q\mathrm{\Theta }{A}_2^q\right)=\left(A_1^p\underset{\alpha }{\#}{A}_1^q\right)\mathrm{\Theta }\left(A_2^p\underset{\alpha }{\#}{A}_2^q\right)\hfill \\ =A_1^{(1-\alpha )p+\alpha q}\mathrm{\Theta }{A}_2^{(1-\alpha )p+\alpha q}={(A_1\mathrm{\Theta }{A}_2)}^{(1-\alpha )p+\alpha q}\text{.}\hfill \end{array}$$

Let A_i > 0(1 ⩽ i ⩽ k,k ⩾ 2) be n × n compatible partitioned matrices. Then

The proof is straightforward by using Theorem 2.2 and induction on k.

Let A_i> and B_i > 0(1 ⩽ i ⩽ k,k ⩾ 2) be n × n compatible partitioned matrices. Then

The proof follows immediately by induction on k.

Let A_i > 0 and B_i > 0(1 ⩽ i ⩽ k,k ⩾ 2) be n × n compatible partitioned matrices and let f(t) be a non-negative operator monotone function on [0,∞) such that $f\left({\prod }_{i=1}^k\mathrm{\Theta }{D}_i\right)={\prod }_{i=1}^k\mathrm{\Theta }f(D_i)$ for any matrices D_i (1 ⩽ i ⩽ k,k ⩾ 2). Then

The proof is straightforward by induction on k and Eqs. 1-13,1-15 and 1-16.

Let $A_i\text{,}{B}_i\in H_{n_i}^{+}(i=1\text{,}2\text{,}\dots \text{,}k)$ and 0 < λ < 1. Then the map φ from $H_{n_1}^{+}\times \cdots \times H_{n_k}^{+}$ to H_m is said to be :

1. Convex if

   (3-1)
   $$\phi (\lambda A_1+(1-\lambda )B_1\text{,}\dots \text{,}\lambda A_k+(1-\lambda )B_k)⩽\lambda \phi (A_1\text{,}\dots \text{,}{A}_k)+(1-\lambda )\phi (B_1\text{,}\dots \text{,}{B}_k)\text{.}$$

2. Concave if the map (A₁,…,A_k) ↦  − φ(A₁,…,A_k) is convex

3. Affine if

   (3-2)
   $$\phi (\lambda A_1+(1-\lambda )B_1\text{,}\dots \text{,}\lambda A_k+(1-\lambda )B_k)=\lambda \phi (A_1\text{,}\dots \text{,}{A}_k)+(1-\lambda )\phi (B_1\text{,}\dots \text{,}{B}_k)\text{.}$$

Let f be a real valued continuous function . Then

1. f is Supermultiplicative if

   (3-3)
   $$f(\mathit{xy})⩾f(x)f(y)\text{.}$$

2. f is Submultiplicative if

   (3-4)
   $$f(\mathit{xy})⩽f(x)f(y)\text{.}$$

Let φ be a normalized positive linear map and σbe an operator mean which has the representation function f which is not affine (f is an operator-monotone on (0,∞)). If A and B are positive definite matrices, then the following statements are equivalent:

It suffices to show that (ii) implies (i). Consider the map ψ defined by

let φ be a positive linear map and let A and B be positive definite matrices. Then

Consider the map ψ defined by $$\psi (X)=\phi {(B)}^{-1/2}\phi (B^{1/2}{\mathit{XB}}^{1/2})\phi {(B)}^{-1/2}\text{.}$$ By a nice technique in the proof of Lemma 3.3, set f(t) = t^α for any real number α, we get 3-8.

The following results as in Theorems 3.5 and 3.6 are referring to Ando (1979).

Let $A\in H_n^{+}$ . Then the map

1. A ↦ A^p is concave if 0 < p ⩽ 1 and is convex if 1 ⩽ p ⩽ 2 or −1 ⩽ p < 0.

2. A ↦ log[A] is concave, while the map A ↦  Alog[A] is convex.

Let φ be a normalized positive linear map from H_n to H_m and A > 0. Then

Let $A_i\in H_n^{+}(1⩽i⩽k\text{,}k⩾2)$ be commutative compatible partitioned matrices. Then

The proof is straightforward by setting $\phi (A_1\text{,}\dots \text{,}{A}_k)={\prod }_{i=1}^k\mathrm{\Theta }{A}_i$ in Theorem 3.6.

Let $A_i\in H_n^{+}(1⩽i⩽k\text{,}k⩾2)$ be commutative compatible partitioned matrices. Then

The proof follows immediately by applying 1-11 and 1-12 on Theorem 3.7.

Let A_i and $B_i\in H_n^{+}$ , (1 ⩽ i ⩽ k,k ⩾ 2) be compatible partitioned matrices. Let σ be an operator mean with supermultiplicative representing function f. Then

(i) Set $X_i=A_i^{-1/2}{B}_i{A}_i^{-1/2}(i=1\text{,}2\text{,}\dots \text{,}k)$ , then it follows from supermultiplicative of f that

Let φ be a positive linear map, then for any compatible partitioned matrices A_i and $B_i\in H_n^{+}(i=1\text{,}2)$

It follows by replacing A by A₁ΘB₁ and B by A₂ ΘB₂ in Theorem 3.4.

Let φ be a positive linear map, then for any compatible partitioned matrices A_i and $B_i\in H_n^{+}(1⩽i⩽k\text{,}k⩾2)$

The proof is straightforward by using Corollary 3.10 and induction on k.

Let A_i and $B_i\in H_n^{+}(i=1\text{,}2)$ be compatible partitioned matrices. Then

Due to Corollary 3.10, Lemma 2.1 and using 1-11 and 1-12, then there is a normalized positive linear map φ such that $$\begin{array}{c}(A_1\underset{\alpha }{\#}{A}_2)\ast (B_1\underset{\alpha }{\#}{B}_2)=\phi ((A_1\mathrm{\Theta }{B}_1)\underset{\alpha }{\#}(A_2\mathrm{\Theta }{B}_2))\hfill \\ =\phi ((A_1\underset{\alpha }{\#}{A}_2)\mathrm{\Theta }(B_1\mathrm{\Theta }{B}_2))\hfill \\ ⩽\phi (A_1\mathrm{\Theta }{B}_1)\underset{\alpha }{\#}\phi (A_2\mathrm{\Theta }{B}_2)=(A_1\ast B_1)\underset{\alpha }{\#}(A_2\ast B_2)\text{.}\hfill \end{array}$$

Let A_i and $B_i\in H_n^{+}(1⩽i⩽k\text{,}k⩾2)$ be compatible partitioned matrices. Then

The proof is straightforward by using Corollary 3.12 and induction on k.

Let $A_i\in H_n^{+}(1⩽i⩽k\text{,}k⩾2)$ be compatible partitioned matrices. Then for −∞ < p,q < ∞,

Due to Theorem 2.3, Theorem 3.4 and using 1-11 and 1-12, then there is a normalized positive linear map φ such that $$\begin{array}{c}\phi \left[\left({\displaystyle \prod _{i=1}^k}\mathrm{\Theta }{A}_i^p\right)\#\left({\displaystyle \prod _{i=1}^k}\mathrm{\Theta }{A}_i^q\right)\right]=\phi \left({\displaystyle \prod _{i=1}^k}\mathrm{\Theta }{A}_i^{(p+q)/2}\right)\hfill \\ ⩽\phi \left({\displaystyle \prod _{i=1}^k}\mathrm{\Theta }{A}_i^p\right)\#\phi \left({\displaystyle \prod _{i=1}^k}\mathrm{\Theta }{A}_i^q\right)=\left({\displaystyle \prod _{i=1}^k}\ast A_i^p\right)\#\left({\displaystyle \prod _{i=1}^k}\ast A_i^q\right)\text{.}\hfill \end{array}$$

Let A and $B\in H_n^{+}$ be compatible partitioned matrices such that A∗B = B∗A. Then

Since A∗B = B∗A, then $$(A\ast B)\#(B\ast A)=(A\ast B)\#(A\ast B)=(A\ast B)\text{.}$$ Since A#B = B#A and from Corollary 3.12, then we have $$(A\ast B)\#(B\ast A)=(A\ast B)⩾(A\#B)\ast (B\#A)=(A\#B)\ast (A\#B)\text{.}$$

Let A_i and $B_i\in H_{n_i}^{+}(1⩽i⩽k)$ be compatible partitioned matrices and let φ_i be a concave map from $H_{n_i}^{+}$ to $H_{m_i}^{+}(1⩽i⩽k)$ . Then the map

It suffices to show the convexity when λ = 1/2. Since the map under consideration is continuous, then $$\begin{array}{c}{\displaystyle \prod _{i=1}^k}\mathrm{\Theta }{\phi }_i{(\lambda A_i+(1-\lambda )B_i)}^{-1}={\displaystyle \prod _{i=1}^k}\mathrm{\Theta }{\phi }_i{\left(\frac{1}{2}(A_i+B_i)\right)}^{-1}\hfill \\ ⩽{\displaystyle \prod _{i=1}^k}\mathrm{\Theta }{\left(\frac{1}{2}\left\{{\phi }_i(A_i)+{\phi }_i(B_i)\right\}\right)}^{-1}(\mathrm{Concavity}\mathrm{of}{\phi }_i)\hfill \\ ⩽{\displaystyle \prod _{i=1}^k}\mathrm{\Theta }{({\phi }_i(A_i)\#{\phi }_i(B_i))}^{-1}={\displaystyle \prod _{i=1}^k}\mathrm{\Theta }({\phi }_i{(A_i)}^{-1}\#{\phi }_i{(B_i)}^{-1})\hfill \\ =\left\{{\displaystyle \prod _{i=1}^k}\mathrm{\Theta }{\phi }_i{(A_i)}^{-1}\right\}\#\left\{{\displaystyle \prod _{i=1}^k}\mathrm{\Theta }{\phi }_i{(B_i)}^{-1}\right\}(\mathit{Theorem}(2.4)\hfill \\ ⩽\frac{1}{2}\left\{{\displaystyle \prod _{i=1}^k}\mathrm{\Theta }{\phi }_i{(A_i)}^{-1}+{\displaystyle \prod _{i=1}^k}\mathrm{\Theta }{\phi }_i{(B_i)}^{-1}\right\}\text{.}\hfill \end{array}$$

Let $A_i\in H_{n_i}^{+}(1⩽i⩽k)$ be compatible partitioned matrices and let 0 ⩽ p_i ⩽ 1 (1 ⩽ i ⩽ k). Then the map

The proof is straightforward by applying Theorems 3.16 and 3.5.

Let $A_i\in H_{n_i}^{+}(1⩽i⩽k)$ be compatible partitioned matrices and let 0 ⩽ p_i ⩽ 1 (1 ⩽ i ⩽ k) such that ${\sum }_{i=1}^k{p}_i⩽1$ . Then the map

The proof is by induction on k. If k = 1, then the result is true by Theorem 3.5. Suppose that Eq. 3-35 is true for the case k − 1. If p_k = 1, then p_i = 0 (1 ⩽ i ⩽ k − 1) and the map becomes $(A_1\text{,}\dots \text{,}{A}_k)\mapsto I_1\mathrm{\Theta }\cdots \mathrm{\Theta }{I}_{n_{k-1}}\mathrm{\Theta }{A}_k$ , which is concave. If p_k = 0, then the map becomes $$(A_1\text{,}\dots \text{,}{A}_k)\mapsto A_1^{p_1}\mathrm{\Theta }\cdots \mathrm{\Theta }{A}_k^{p_{k-1}}\mathrm{\Theta }{I}_{n_k}\text{,}$$ which is concave. Now suppose 0 < p_k < 1. Then the map $$(A_1\text{,}\dots \text{,}{A}_{k-1})\mapsto {\displaystyle \prod _{i=1}^{k-1}}\mathrm{\Theta }{A}_i^{p_i/(1-p_k)}$$ is concave by the induction assumption. Now with $f(\lambda )={\lambda }^{p_k}$ , the map $$(A_1\text{,}\dots \text{,}{A}_k)\mapsto {\displaystyle \prod _{i=1}^k}\mathrm{\Theta }{A}_i^{p_i}$$ is concave.