1 Introduction

Inequalities play an important role in mathematics, engineering and science. In operator theory, several operator inequalities appear related to operator monotone functions in the context of Kubo-Ando theory on means [16]. The main purpose of this note is to survey some results on means defined on indefinite inner product spaces, shortly named indefinite means inequalities and to prove the equivalence of relevant inequalities in this context. We will restrict our attention to the finite dimensional case, but the investigation in the infinite dimensional set up also deserves attention. We start recalling the concept of mean. A continuous function \(M\!:(0,\infty )\times (0,\infty )\rightarrow (0,\infty )\) is a mean of two positive numbers if it satisfies the following conditions:

  1. i.

    \(M(a,a)=a,\) for every a;

  2. ii.

    \(M(a,b)=M(b,a)\), for every a, b;

  3. iii.

    \(\min \{a,b\}\le M(a,b)\le \max \{a,b\}\);

  4. iv.

    M(a, b) is monotone increasing in a and b;

  5. v.

    \(M(\alpha \, a,\alpha \,b )=\alpha \,M(a,\,b)\), for \(\alpha >0.\)

For two positive numbers a, b,  the well known arithmetic mean, geometric mean, harmonic mean are defined as

$$\begin{aligned} \frac{a+b}{2},\qquad \sqrt{a\,b},\qquad \frac{2a\,b}{a+b}, \end{aligned}$$

respectively. These means were already among the family of means praised and studied by the ancient Pithagoreans six centuries before Christ.

An archetypal procedure of mathematicians consists of generalizing concepts. Inequalities for positive numbers could have plausible extensions to positive definite matrices. In this perspective, researchers generalized means of positive numbers to means of positive (semi)definite Hermitian matrices A, B or Hilbert space operators. These include the most famous Kubo-Ando means [16] and the quasi-arithmetic means, for example, characterizations concerning these means were given in [21]. The matrix versions of the \(\alpha \)-weighted arithmetic and harmonic means for \(\alpha \in [0,1]\) are

$$\begin{aligned} A \,\nabla _\alpha \, B \,= \, \alpha \,A+(1-\alpha )\,B\qquad \hbox {and} \qquad A \,!_\alpha \, B \,= \,\left( \alpha A^{-1} + (1-\alpha )\, B^{-1} \right) ^{-1}, \end{aligned}$$

respectively. We will be specially concerned with the \(\alpha \)-weighted geometric mean of matrices, originally called \(\alpha \)-power mean by Ando and Hiai [3], previously defined by Pusz and Woronowicz [22] for \(\alpha =\frac{1}{2}\) and in such case called geometric mean.

We use the notation \(A>0\) (\(A\ge 0\)) if \(A\in {{\mathbb {C}}}^{n\times n}\) is a positive (semi)definite Hermitian matrix; \(A \ge B\) if \(A, B\in {{\mathbb {C}}}^{n\times n}\) are Hermitian matrices and \(A-B\) is positive semidefinite. The \(\alpha \)-weighted geometric mean of \(A,B \ge 0\) for \(\alpha \in [0,1]\) is given by

$$\begin{aligned} A \, \sharp _{\alpha }\, B\,=\, A^{\,\frac{1}{2}} \big ( A^{-\frac{1}{2}} B\, A^{-\frac{1}{2}}\big )^{\alpha } A^{\,\frac{1}{2}} \end{aligned}$$

when A is invertible and it is extended to any non-invertible A by continuity as follows:

$$\begin{aligned} A \, \sharp _{\alpha }\, B\, =\, \lim _{\epsilon \rightarrow 0^+}\, (A+\epsilon I) \, \sharp _{\alpha } \,B. \end{aligned}$$

In particular, \(\sharp =\sharp _{1/2}\) denotes the geometric mean. If \(AB=BA\), then we have \(A \, \sharp _{\alpha }\, B\,=\,A^{1-\alpha } B^{\alpha }.\) In general, \(A\,\sharp _{1-\alpha }\, B=B\,\sharp _{\alpha }\, A\) and \((A,B)\mapsto A \, \sharp _{\alpha }\, B\) is jointly monotone both in A and B, as a consequence of the famous Löwner–Heinz inequality.

Löwner–Heinz inequality (L-H):

$$\begin{aligned} A \ge B \ge 0 \quad implies \quad A^\alpha \ge B^\alpha \quad f\!or \ \ \alpha \in [0,1]. \end{aligned}$$

The original proof is due to Löwner [17] and it used an integral representation for operator monotone functions. Heinz [14] gave an alternative proof. A celebrated development of Löwner–Heinz inequality established in 1987 by T. Furuta [12] is the next order preserving operator inequality, which was motivated by a previous conjecture by Chan and Kwong [10].

Furuta inequality (F):

If \(A\ge B\ge 0\), then for each \(r\ge 0\)

$$\begin{aligned} A^\frac{p+r}{q}\ge \left( A^{\frac{r}{2}} B^p A^{\frac{r}{2}}\right) ^{\frac{1}{q}}\quad \hbox {and} \quad \left( B^{\frac{r}{2}} A^p B^{\frac{r}{2}}\right) ^{\frac{1}{q}}\ge B^\frac{p+r}{q} \end{aligned}$$
(1)

hold for \(p\ge 0\) and \(q\ge 1\) with \((1+r)q\ge p+r.\)

The case of p, q, r all equal to 2 in (1) affirmatively answers Chan and Kwong’s conjecture:

$$\begin{aligned} A\ge B\ge 0\quad \Rightarrow \quad A^2\ge \left( A\, B^2 A\right) ^{\frac{1}{2}}. \end{aligned}$$

Furuta and many other researchers refined and generalized (1) and applied these results to produce new inequalities [13].

Ando and Hiai proved a log-majorization result [3, Theorem 2.1], concerning the eigenvalues of the \(\alpha \)-weighted geometric mean, which is known as Ando–Hiai inequality and it can be rewritten as follows.

Ando–Hiai inequality (A-H):

For \(A,B\ge 0\) and \(\alpha \in [0,1]\), if \(A\sharp _\alpha B \le I\) holds, then \(A^r\sharp _\alpha B^r\le I\) holds for \(r\ge 1\).

This note is organized as follows. In Sect. 2, we present inequalities obtained in an indefinite inner product space context and for simplicity they will be called indefinite inequalities. For the general theory of these spaces we refer the reader to [4]. In Sect. 3, we prove that the indefinite Ando–Hiai inequality and the indefinite Furuta inequality are equivalent.

2 Indefinite inequalities

Let J be a non-trivial involutive Hermitian matrix, that is, \(J^*=J\), \(J^2=I\) and \(J \ne I\). Consider \({{\mathbb {C}}}^n\) equiped with the indefinite inner product induced by J, that is, defined by

$$\begin{aligned}{}[x,y]=y^*J x, \quad x,y\in {{\mathbb {C}}}^n. \end{aligned}$$

The J-adjoint of \(A\in {{\mathbb {C}}}^{n\times n}\) is the matrix \(A^{[*]}\in {{\mathbb {C}}}^{n\times n}\), satisfying \([Ax,y]=[ x, A^{[*]} y]\) for all \(x,y\in {{\mathbb {C}}}^n,\) or equivalently, \(A^{[*]}=JA^*J.\) A matrix \(A\in {{\mathbb {C}}}^{n\times n}\) is said to be J-Hermitian, or J-selfadjoint, if \(A=A^{[*]}\), that is, if JA is Hermitian. It is well known that the eigenvalues of such matrices, which appear in several problems of relativistic quantum mechanics and quantum physics [5, 9], may not be real, nevertheless they occur in pairs of complex conjugate numbers. We denote the spectrum of A by \(\sigma (A)\).

For \(A,B\in {{\mathbb {C}}}^{n\times n}\) J-selfadjoint matrices we define the J-order \(A\ge ^J B\) as

$$\begin{aligned}{}[A x,x]\ge ^J [B x,x], \qquad x\in {{\mathbb {C}}}^n, \end{aligned}$$

or equivalently, \(J(A-B)\) is positive semidefinite.

If \(I\ge ^J A\), then all the eigenvalues of A are real, since \(I-A\) is the product of the Hermitian matrix J by a positive semidefinite matrix. A matrix \(A\in {{\mathbb {C}}}^{n\times n}\) is said to be a J-contraction if \(I\ge ^J A^{[*]}A\), equivalently, \(J\ge ^J A^*J A\) and in this case, by a Theorem of Potapov Ginzburg [4], all the eigenvalues of \(A^{[*]} A\) are nonnegative.

Following T. Ando [2], let us consider the following class of matrices:

$$\begin{aligned} {\mathcal U} \,= \,\big \{A\in {{\mathbb {C}}}^{n\times n}\!: \ I\ge ^J A \, \ \wedge \ \, \sigma (A)\subseteq [\,0,\infty )\big \}. \end{aligned}$$

If A is a J-contraction, then \(A^{[*]} A\in \, \mathcal U\). The class  \(\mathcal U\) is not convex [2]. The spectral inclusion in class  \(\mathcal U\) can be replaced by an operator inequality [2, Theorem 3.1], that is,

$$\begin{aligned} {\mathcal U} \,= \,\big \{A\in {{\mathbb {C}}}^{n\times n}\!: \ I\ge ^J A\ge ^J A^2\big \} \end{aligned}$$

and, consequently,

$$\begin{aligned} A,B\in \, \mathcal U \quad \Rightarrow \quad ABA \in \,\mathcal U. \end{aligned}$$

[2, Theorem 3.5]. For simplicity, let \({\mathcal U}^*\) be the set of invertible elements in \(\mathcal U\). Then

$$\begin{aligned} {\mathcal U}^* = \, \big \{A\in {{\mathbb {C}}}^{n\times n}\!: \ I\ge ^J A \, \ \wedge \ \, \sigma (A)\subseteq (0,\infty )\big \} \end{aligned}$$
(2)

or equivalently,

$$\begin{aligned} {\mathcal U}^* = \, \big \{A\in {{\mathbb {C}}}^{n\times n}\!: \ A \ \,\hbox {is invertible} \,\ \wedge \ \, A^{-1}\ge ^J I\ge ^J A\big \} \end{aligned}$$

(see [23, Lemma 2.3] and [2, Corollary 3.3]).

A real valued continuous function f defined on a real (finite or infinite) interval \((\alpha ,\beta )\) is an operator monotone function if \(A\ge B\) implies \(f(A)\ge f(B)\) for any pair of matrices \(A,B \in {{\mathbb {C}}}^{n\times n}\) with spectra in \((\alpha ,\beta )\), for all \(n \in {\mathbb {N}}\). If all eigenvalues of a J-selfadjoint matrix A are real and \(\sigma (A)\subseteq (\alpha ,\beta )\), then for any operator monotone function f on \((\alpha ,\beta )\), we can define f(A) by the Riesz-Dunford integral

$$\begin{aligned} f(A)=\frac{1}{2\pi \,i}\int _\Gamma f(\xi )(\xi \,I-A)^{-1}d\xi , \end{aligned}$$
(3)

where \(\Gamma \) is a closed rectifiable contour in the domain of analytic continuation of f,  surrounding positively the spectrum of A. Further, f(A) is J-selfadjoint.

If A is Hermitian, this integral produces the same matrix as that defined by the usual functional calculus for a Hermitian matrix.

Ando [1, Theorem 4] obtained the following result concerning operator monotone functions.

Theorem 2.1

Let \(A,B\in {{\mathbb {C}}}^{n\times n}\) be J-selfadjoint with \(\sigma (A),\sigma (B)\subseteq (\alpha ,\beta ).\) Then

$$\begin{aligned} A\ge ^J B \quad \Rightarrow \quad f(A)\ge ^J f(B) \end{aligned}$$

for any operator monotone function f on \((\alpha ,\beta ).\)

By Theorem 2.1, due to the operator monotonicity on \((0,\infty )\) of \(f(t)=t^\alpha \), \(\alpha \in [0,1]\), stated by (L-H), if \(A,B\in {{\mathbb {C}}}^{n\times n}\) are J-selfadjoint with \(\sigma (A),\sigma (B)\subseteq (0,\infty )\), then

$$\begin{aligned} A\ge ^JB \quad \Rightarrow \quad A^{\alpha }\ge ^J B^{\alpha } \quad \hbox {for} \quad \alpha \in [0,1]. \end{aligned}$$

If \(\sigma (A)\subseteq [\,0,\infty ),\) some conditions must be imposed so that the integral (3) is convergent, namely \(I\ge ^J A\); in particular, for \(f(t)=t^\frac{1}{2}\) the integral gives a \(J-\)selfadjoint square root of A with non-negative eigenvalues [1, Theorem 4]. In this context, following Ando [1, Theorem 6] for the case \(\alpha =\frac{1}{2}\), Sano [23, Theorem 2.4] presented the following result, which is named the indefinite Löwner-Heinz inequality.

Indefinite Löwner-Heinz inequality (IL-H):

For selfadjoint matrices \(A,B\in {{\mathbb {C}}}^{n\times n}\), such that \(\sigma (A),\sigma (B)\subseteq [\,0,\infty )\),

$$\begin{aligned} I\ge ^J A\ge ^J B \quad \Rightarrow \quad I\ge ^J A^{\alpha }\ge ^J B^{\alpha } \quad \hbox {for} \ \ \alpha \in [0,1]. \end{aligned}$$

After this result, several indefinite inequalities for J-selfadjoint matrices and J-contractions have been studied (see e.g. [6,7,8, 18,19,20]), some of them are stated in the next section.

3 Ando–Hiai and Furuta inequalities of indefinite type are equivalent

In the sequel, let \( \alpha \in [0, 1]\). The \(\alpha \)-weigthed geometric mean can be naturally extended to J-selfadjoint matrices \(A,B\in {{\mathbb {C}}}^{n\times n}\) with \(\sigma (A), \sigma (A^{-1}B)\subseteq (0, \infty )\), following the previous definition:

$$\begin{aligned} A\,\sharp _\alpha \, B \ = \ A^{\frac{1}{2}}\big (A^{-\frac{1}{2}}B A^{-\frac{1}{2}}\big )^\alpha A^{\frac{1}{2}}, \end{aligned}$$

being the subscript \(\alpha \) in \(\sharp _\alpha \) omitted when \(\alpha =\frac{1}{2}\). In this case, \(A\,\sharp _\alpha \, B \) is a J-selfadjoint matrix too. Moreover, \(\sharp _{\alpha }\) is positively homogeneous, that is, \((\mu A)\,\sharp _{\alpha }\, (\mu B)=\mu \,(A\sharp _{\alpha } B)\) for any \(\mu >0\) and \((A\,\sharp _{\alpha }\, B)^{-1}=A^{-1}\sharp _\alpha \, B^{-1}\).

In general, the J-mean of A and B under the previous conditions could be defined for any normalized (i.e. \(f(1)=1\)) operator monotone function f on \((0, \infty )\) by

$$\begin{aligned} A\,\sigma _f\, B \ = \ A\,f(A^{-1}B) \ = \ A^{\frac{1}{2}}\,f\big (A^{-\frac{1}{2}}B\, A^{-\frac{1}{2}}\big )\, A^{\frac{1}{2}} \end{aligned}$$

(see [18]) in paralell to Kubo-Ando means of positive operators. Our attention will be restricted to the case \(f(t)=t^\alpha \), \(\alpha \in [0, 1]\).

It should be remarked that the J-selfadjoint matrix \(A\,\sharp _\alpha \, B\) may not have real eigenvalues. Indeed, for example, consider the J-selfadjoint matrices

$$\begin{aligned} A=\left[ \begin{array}{cc} 5 &{} -3 \\ 3 &{} -1 \\ \end{array} \right] \qquad \hbox {and} \qquad B=\left[ \begin{array}{cc} -1 &{} -3 \\ 3 &{} 5 \\ \end{array} \right] , \qquad \hbox {for} \qquad J=\left[ \begin{array}{cc} 1 &{} 0 \\ 0 &{} -1 \\ \end{array} \right] , \end{aligned}$$

with \(\sigma (A)=\sigma (B)=\{2\}\subseteq (0,\infty )\). By simple computations, \(J(A-B)=6\, I\) and

$$\begin{aligned} \sigma (A^{-1}B)=\left\{ \frac{11 + 3\sqrt{13}}{2}, \frac{11 - 3\sqrt{13}}{2}\right\} \subseteq (0,\infty ). \end{aligned}$$

Nevertheless, the J-selfadjoint matrix

$$\begin{aligned} A\,\sharp \, B=\frac{1}{\sqrt{13}}\left[ \begin{array}{cc} 4 &{} -6 \\ 6 &{} 4 \\ \end{array} \right] \end{aligned}$$

has non-real eigenvalues, namely

$$\begin{aligned} \sigma (A\,\sharp \, B)= \left\{ \frac{4+6 i}{\sqrt{13}},\frac{4-6 i}{\sqrt{13}}\right\} . \end{aligned}$$

In fact, \(A\ge ^J B\) is satisfied, but A, B are not in class \(\,{\mathcal U}^*\), because the matrices

$$\begin{aligned} J(I-A)=\left[ \begin{array}{cc} -4 &{} 3 \\ 3 &{} -2 \\ \end{array} \right] \quad \hbox { and } \quad J(I-B)=\left[ \begin{array}{cc} 2 &{} 3 \\ 3 &{} 4 \\ \end{array} \right] \end{aligned}$$

are not positive semidefinite, as their eigenvalues are \(-3\pm \sqrt{10}\) and \(3\pm \sqrt{10}\), respectively.

The following basic results will be used throughout.

Lemma 3.1

[6]. Let \(A, B\in {{\mathbb {C}}}^{n\times n}\) be J-selfadjoint. Then \(X^{[*]}AX\ge ^J X^{[*]}BX\) for all \(X\in {{\mathbb {C}}}^{n\times n}\) if and only if \(A\ge ^J B\).

Lemma 3.2

[23]. If \(A \in \, \mathcal U\), then \(I\ge ^J A^{\lambda }\) for all \(\lambda >0.\)

From Theorem 2.1 and the fact that \(f(t)=-\frac{1}{t}\) is an operator monotone function on \((0,\infty ),\) the following holds.

Lemma 3.3

For \(A,B\in {{\mathbb {C}}}^{n\times n}\) J-selfadjoint with positive spectra, if \(A\ge ^J B,\) then \(B^{-1}\ge ^J A^{-1}.\)

Proposition 3.4

[18]. Let \(A, B \in {{\mathbb {C}}}^{n\times n}\) be J-selfadjoint with positive spectra. If \(A\ge ^J B\), then \(A\,\sharp _\alpha \, B\,=\,B\,\sharp _{1-\alpha }\, A\) for any \(\alpha \in [0,1].\)

If \(A\ge ^J B\) with \(\sigma (A),\,\sigma (B) \subseteq (0, \infty )\), then by Lemma 3.1 we have

$$\begin{aligned} I\, \ge ^J A^{-\frac{1}{2}}B\, A^{-\frac{1}{2}}\qquad \hbox {and} \qquad B^{-\frac{1}{2}}A\, B^{-\frac{1}{2}} \ge ^J \, I, \end{aligned}$$

that is, \(B^{\frac{1}{2}} A^{-\frac{1}{2}}\) and \( A^{\frac{1}{2}} B^{-\frac{1}{2}}\) are invertible J-contractions, so that \(A^{-\frac{1}{2}}B A^{-\frac{1}{2}}\) and \(B^{-\frac{1}{2}}A B^{-\frac{1}{2}}\) have positive spectra. By Theorem 2.1 and Lemma 3.1, we get

$$\begin{aligned} A \,=\, A^{\frac{1}{2}}\, I\, A^{\frac{1}{2}} \ge ^J A^{\frac{1}{2}}\big (A^{-\frac{1}{2}}B\, A^{-\frac{1}{2}}\big )^\alpha A^{\frac{1}{2}} = \, A\,\sharp _\alpha \, B \end{aligned}$$
(4)

and, using also Proposition 3.4,

$$\begin{aligned} A\,\sharp _\alpha \, B \, = \, B\,\sharp _{1-\alpha }\, A \, = \, B^{\frac{1}{2}}\big (B^{-\frac{1}{2}}A\, B^{-\frac{1}{2}}\big )^{1-\alpha } B^{\frac{1}{2}} \ge ^J B^{\frac{1}{2}}\, I\, B^{\frac{1}{2}} =\, B. \end{aligned}$$

In this case, we obtained \(A \ge ^J A\,\sharp _\alpha \, B \ge ^J B,\) that is, \(\sharp _\alpha \) is a mean with respect to the J-order. Analogously, if \(B\ge ^J A\) and \(\sigma (A),\, \sigma (B)\subseteq (0, \infty )\), then \(B \ge ^J A\,\sharp _\alpha \, B \ge ^J A\).

Now, if A, B are in class \(\,\mathcal U^*\), we can conclude the following, which ensures that the J-selfadjoint matrix \(A\,\sharp _\alpha \, B\) has positive real eigenvalues too.

Proposition 3.5

Let \(A,B \in \, \mathcal U^*\) and \(A\ge ^J B\), then \(A\,\sharp _\alpha \, B \in \, {\mathcal U}^*\) for any \(\alpha \in [0,1]\).

Proof

If \(A\in \, {\mathcal U}^*\), then \(I \ge ^J A \ge ^J A\sharp _\alpha B\), having in mind (4), which implies that

$$\begin{aligned} W=\big (A^{-\frac{1}{2}}B A^{-\frac{1}{2}}\big )^\frac{\alpha }{2} A^\frac{1}{2} \end{aligned}$$

is a J-contraction, that is, \(I \ge ^J W^{[*]} W\). Hence, \(W^{[*]} W =A\,\sharp _\alpha \, B\), which is invertible, has positive spectra and we conclude that \(A\,\sharp _\alpha \, B \in \, {\mathcal U}^*\). \(\square \)

By Lemma 3.1, Theorem 2.1 (and Proposition 3.4), we easily see that \(\sharp _\alpha \) satifies a monotonicity property in the first (and second) variables with respect to the J-order.

Proposition 3.6

Let \(A,B,C\in {{\mathbb {C}}}^{n\times n}\) be J-selfadjoint matrices with positive spectra.

(a):

If \(A\ge ^J B\ge ^J C\), then \(A\,\sharp _\alpha \, B \ge ^J A\,\sharp _\alpha \, C\).

(b):

If \( A\ge ^J C\ge ^J B\), then \(A\,\sharp _\alpha \, B \ge ^J C\,\sharp _\alpha \, B\).

Motivated by (IL-H), Furuta inequality of indefinite type was established as follows (see [23, Theorem 3.4] and [6, Theorem 2.1]).

Indefinite Furuta inequality (IF):

Let \(A,B\in {{\mathbb {C}}}^{n\times n}\) be J-selfadjoint with nonnegative spectra. If \(\mu I\ge ^J A\ge ^J B\) (or \(A\ge ^J B\ge ^J \mu I\)) for some \(\mu >0\), then for each \(r\ge 0\)

$$\begin{aligned}{} & {} A^\frac{p+r}{q}\ge ^J \left( A^{\frac{r}{2}} B^p A^{\frac{r}{2}}\right) ^{\frac{1}{q}}\qquad \hbox {and} \qquad \nonumber \\{} & {} \left( B^{\frac{r}{2}} A^p B^{\frac{r}{2}}\right) ^{\frac{1}{q}}\ge ^J B^\frac{p+r}{q} \end{aligned}$$
(5)

hold for \(p\ge 0\) and \(q\ge 1\) with \((1+r)q\ge p+r.\)

We notice that (IL-H) is recovered from (IF) when \(r=0\). The essential part of the indefinite Furuta inequality (EIF) is the case \(q=\frac{p+r}{1+r}\) and \(p\ge 1\), because the case \(q>\frac{p+r}{1+r}\) then follows from the essential one, using (IL-H). Without loss of generality, we may suppose \(\mu =1\), otherwise we replace A, B by \(\frac{1}{\mu }A, \frac{1}{\mu }B\), respectively, and we may consider A, B invertible by [23, Proposition 3.3], such that \(I\ge ^J A\ge ^J B\) (if \(A\ge ^J B\ge ^J I\), the result follows from \(I\ge ^J A^{-1}\ge ^J B^{-1}\), using Lemma 3.3). Hence, this essential part is stated for \(A,B\in {{\mathbb {C}}}^{n\times n}\) J-selfadjoint with positive eigenvalues as

$$\begin{aligned} A \ge ^J B \quad \Rightarrow \quad A^{1+r}\ge ^J \left( A^{\frac{r}{2}}\,B^p\,A^{\frac{r}{2}}\right) ^{\frac{1+r}{p+r}} \quad \hbox { and } \quad \left( B^{\frac{r}{2}}\,A^p\,B^{\frac{r}{2}}\right) ^{\frac{1+r}{p+r}}\ge B^{1+r} \end{aligned}$$

for all \(p\ge 1\) and \(r\ge 0\), which can be written using the \(\alpha \)-weighted geometric mean as follows.

Corollary 3.7

(EIF). If \(A, B \in \, \mathcal U^*\), then \(A \ge ^J B\) implies

$$\begin{aligned} A = A^{-r}\,\sharp _{s_1}\,\ A^p \ge ^J A^{-r}\,\sharp _{s_1}\ B^p \quad \ \hbox {and}\ \quad B^{-r}\sharp _{s_1}\, A^p \ \ge ^J \ B^{-r}\sharp _{s_1}\, B^p = B \end{aligned}$$

for all \(p\ge 1\) and \(r\ge 0\), with \(s_1=\frac{1+r}{p+r}\).

For \(A,B\in {{\mathbb {C}}}^{n\times n}\) J-selfadjoint with positive spectra, the J-chaotic order is defined by \({\textrm{Log}}(A)\ge ^J{\textrm{Log}}(B),\) where \({\textrm{Log}}\) denotes the principal branch of the logarithm function. Since \({\textrm{Log}}(t)\) is an operator monotone function on \((0,\infty ),\) by Theorem 2.1 the \(J-\)chaotic order \({\textrm{Log}}(A)\ge ^J{\textrm{Log}}(B)\) is weaker than the usual J-order \(A\ge ^J B.\)

If \(A,B\in \,{\mathcal U}^*\), then \({\textrm{Log}}(A)\ge ^J {\textrm{Log}}(B)\) if and only if \( A^r \,\ge ^J \big (A^{\frac{r}{2}}B^p A^{\frac{r}{2}}\big )^{\frac{r}{p+r}} \) for all \(p,r> 0\) [24, Theorem 7]. We may characterize the J-chaotic order for matrices in class  \({\mathcal U}^*\) as follows, inspired by the corresponding equivalent conditions for the case \(t=0\).

Proposition 3.8

[8]. If \(A,B\in \,{\mathcal U}^*\), the following statements are mutually equivalent:

(a):

\({\textrm{Log}}(A)\ge ^J {\textrm{Log}}(B);\)

(b):

\(B^t \ge ^J A^{-r} \,\sharp _{s_t}\, B^p\)  for \(p\ge t\ge 0\) and \(r \ge 0\);

(c):

\(B^{-r}\,\sharp _{s_t}\, A^{p} \ge ^J A^t\)  for \(p\ge t\ge 0\) and \(r \ge 0\);

where \(s_t=\frac{t+r}{p+r}\).

Since \(A\ge ^J B\) implies \({\textrm{Log}}(A)\ge ^J {\textrm{Log}}(B)\), the next indefinite version of Kamei’s satellite to Furuta inequality [15] readily follows from the implications \(\mathbf (a) \Rightarrow (b)\) and \(\mathbf (a) \Rightarrow (c)\) of Proposition 3.8 in the particular case \(t=1\), thus improving Corollary 3.7.

Corollary 3.9

If \(A,B \in \, {\mathcal U}^*\), such that \(A\ge ^J B\), then

$$\begin{aligned} B^{-r}\, \sharp _{s_1}\, A^p \ \ge ^J \, A \ \ge ^J \, B \ \ge ^J \, A^{-r}\, \sharp _{s_1}\, B^p \end{aligned}$$

for all \(p\ge 1\) and \(r\ge 0\), with \(s_1=\frac{1+r}{p+r}\).

Some related inequalities were given in [25, 26].

Moreover, Matharu et al. [18, Theorem 3.11] proved an indefinite version of the Ando–Hiai inequality, which is written below for the operator monotone function \(f(t)=t^\alpha \), \(t>0\), \(\alpha \in [0, 1]\).

Indefinite Ando–Hiai inequality (IA-H):

Let \(A,B\in {{\mathbb {C}}}^{n\times n}\) be J-selfadjoint with positive spectra, such that \(A\ge ^J \mu I\) and \(B\le ^J \gamma I\) for some \(\mu , \gamma >0\) and \( \alpha \in [0,1]\). Then

$$\begin{aligned} I \ge ^J A\, \sharp _{\alpha }\, B \quad \Rightarrow \quad I \ge ^J A^r \sharp _{\alpha }\, B^r \quad \hbox {for} \ \ r\ge 1. \end{aligned}$$

Fujii and Kamei [11] showed that Ando–Hiai inequality is equivalent to Furuta inequality. Inspired by their results we prove that the indefinite Ando–Hiai inequality and the essential part of the indefinite Furuta inequality in Corollary 3.7 are equivalent results too.

Theorem 3.10

The indefinite Ando–Hiai inequality implies the essential part of the indefinite Furuta inequality.

Proof

Let \(A,B\in \,{\mathcal U}^*\), \(p\ge 1\) and \(r\ge 0\). By Lemma 3.2 and Lemma 3.3, we have

$$\begin{aligned} A^{-r} \ge ^J I \, \ge ^J B^p. \end{aligned}$$
(6)

Suppose that \(A\ge ^J B\) and let \(s_0=\frac{r}{p+r}\). If \(0\le r \le 1\), then \(-A^{-r}\ge ^J -B^{-r}\) by Theorem 2.1 for the operator monotone function \(f(t)=-\frac{1}{t^r}\) on \((0, \infty )\). Then

$$\begin{aligned} B^{-r}\ge ^J A^{-r}\ge ^J B^p \end{aligned}$$

and by Proposition 3.6 we have

$$\begin{aligned} I\,=\,B^{-r}\,\sharp _{s_0}\, B^p \ge ^J A^{-r}\sharp _{s_0}\, B^p \ge ^J B^p\, \sharp _{s_0 }\, B^p = B^p. \end{aligned}$$

Next, let \(r>1\). In this case, from

$$\begin{aligned} B^{-1}\ge ^J A^{-1} \ge ^J I \,\ge ^J B^\frac{p}{r}, \end{aligned}$$

we find

$$\begin{aligned} I\,=\,B^{-1}\sharp _{s_0}\, B^\frac{p}{r}\, \ge ^J A^{-1}\sharp _{s_0}\, B^\frac{p}{r}. \end{aligned}$$

By (IA-H) applied to the J-selfadjoint matrices \(A_1=A^{-1}\) and \(B_1= B^\frac{p}{r}\), which have positive spectra, we conclude that

$$\begin{aligned} I\, \ge ^J A^{-r}\sharp _{s_0}\, B^p. \end{aligned}$$

Again from (6), we obtain

$$\begin{aligned} A^{-r}\,\sharp _{s_0}\, B^p \ge ^J B^p\, \sharp _{s_0}\, B^p = B^p \end{aligned}$$

for \(r>1\).

We conclude that

$$\begin{aligned} I\, \ge ^J A \ge ^J B = I\, \sharp _\frac{1}{p}\, B^p\, \ge ^J \big (A^{-r}\,\sharp _{s_0}\, B^p\big )\, \sharp _\frac{1}{p}\, B^p \end{aligned}$$

for any \(r \ge 0\) and, by Proposition 3.4, considering \(s_1=\frac{1+r}{p+r}\), we have

$$\begin{aligned} \big (A^{-r}\sharp _{s_0}\, B^p\big )\, \sharp _\frac{1}{p}\, B^p= & {} B^p\, \sharp _{1-\frac{1}{p}}\, \big (A^{-r}\sharp _{s_0}\, B^p\big ) \\= & {} B^p\, \sharp _{1-\frac{1}{p}}\, \big (B^p\, \sharp _{1-s_0}\, A^{-r}\big ) \\= & {} B^p\, \sharp _{(1-s_0)\left( 1-\frac{1}{p}\right) }\, A^{-r} \\= & {} B^p\, \sharp _{1-s_1}\, A^{-r} \\= & {} A^{-r}\,\sharp _{s_1}\, B^p. \end{aligned}$$

We have just obtained the first inequality of Corollary 3.7. The second one could be analogously obtained. \(\square \)

Theorem 3.11

The essential part of the indefinite Furuta inequality implies the indefinite Ando–Hiai inequality.

Proof

Let \(A,B\in {{\mathbb {C}}}^{n\times n}\) be J-selfadjoint with positive spectra, such that \(A\ge ^J \mu I\) and \(B\le ^J \gamma I\), for some \(\mu , \gamma >0\). Without loss of generality, let \(\mu =\gamma =1\), otherwise replace A and B by \(\frac{1}{\mu }\,A\) and \(\frac{1}{\gamma }\,B,\) respectively. Suppose that

$$\begin{aligned} I \ge ^J A \sharp _\alpha B, \qquad 0\le \alpha \le 1. \end{aligned}$$
(7)

Let \(r=1+\epsilon \) with \(\epsilon \in [0,1]\) and \(C=A^{-\frac{1}{2}}B A^{-\frac{1}{2}}\). We have \(I \ge ^J C\) and from (7) by Lemma 3.1 we have

$$\begin{aligned} I \ge ^J A^{-1}\ge ^J C^\alpha . \end{aligned}$$

By Proposition 3.6 we obtain

$$\begin{aligned} A^{-1}\,\sharp _{1-\epsilon }\, C \ge ^J C^\alpha \,\sharp _{1-\epsilon }\, C\,=\, C^{1-\epsilon (1-\alpha )}, \end{aligned}$$

which is equivalent to

$$\begin{aligned} C^{\epsilon (1-\alpha )-1} \ge ^J (A^{-1}\,\sharp _{1-\epsilon }\, C)^{-1}=A\,\sharp _{1-\epsilon }\, C^{-1} \end{aligned}$$
(8)

by Lemma 3.3. Applying Corollary 3.7, i.e., the essential part of the indefinite Furuta inequality, to \(A_1=A^{-1}\) and \(B_1= C^\alpha \) which are in class \( {\mathcal U}^*\), we find that

$$\begin{aligned} A^{-1} \ge ^J A^{\epsilon }\,\sharp _{\frac{1+\epsilon }{p+\epsilon }}\, C^{\,\alpha p}, \end{aligned}$$

equivalently,

$$\begin{aligned} I \ge ^J A^{1+\epsilon }\,\sharp _{\frac{1+\epsilon }{p+\epsilon }}\, (A^\frac{1}{2}\, C^{\,\alpha p} A^\frac{1}{2}) \end{aligned}$$

for all \(p \ge 1\). We observe that

$$\begin{aligned} B^{1+\epsilon } = A^\frac{1}{2} C\,A^\frac{1}{2} B^{\epsilon -1} A^\frac{1}{2} C\,A^\frac{1}{2} = A^{\frac{1}{2}} C\,\big (A\,\sharp _{1-\epsilon }\, C^{-1}\big ) C\,A^\frac{1}{2}. \end{aligned}$$
(9)

If we choose \(p=\frac{1+\epsilon (1-\alpha )}{\alpha }\), then \(\alpha =\frac{1+\epsilon }{p+\epsilon }\) and \(p\ge 1\). Clearly, \(A^\epsilon \ge ^J I \ge ^J C^{\,\alpha p}\). Recalling (8), again by Lemma 3.1, we get

$$\begin{aligned} A^{1+\epsilon }&\ge ^J A^\frac{1}{2}\, C^{\,\alpha p} A^\frac{1}{2} \\&=\, A^{\frac{1}{2}}\, C^{1+\epsilon (1-\alpha )} \, A^\frac{1}{2} \\&=\, A^{\frac{1}{2}}\, C\, C^{\epsilon (1-\alpha )-1} C\,A^\frac{1}{2}\\&\ge ^J A^{\frac{1}{2}}\, C\,\big (A\,\sharp _{1-\epsilon }\, C^{-1}\big ) C\,A^\frac{1}{2} \\&=\, B^{1+\epsilon }, \end{aligned}$$

having also in mind (9). By Proposition 3.6, we obtain

$$\begin{aligned} I \ge ^J A^{1+\epsilon }\,\sharp _{\alpha }\, (A^\frac{1}{2}\, C^{\,\alpha p} A^\frac{1}{2}) \ge ^J A^{1+\epsilon }\, \sharp _\alpha \, B ^{1+\epsilon }, \end{aligned}$$

that is,

$$\begin{aligned} I \ge ^J A^r\,\sharp _{\alpha }\, B^r,\qquad r \in [1,2]. \end{aligned}$$

When \(r>2\), there exists \(n\in {\mathbb {N}}\) and \(s\in [1,2]\) such that \(r=2^n s\) and the result follows by sucessive iterations. \(\square \)

Now, we recall the complete form of Furuta inequality [27] of indefinite type [8, Theorem 3.1].

Theorem 3.12

Let \(A,B\in {{\mathbb {C}}}^{n\times n}\) be J-selfadjoint with nonnegative eigen-values and \(\mu \, I\ge ^J A\ge ^J B\) (or \( A\ge ^J B\ge ^J \mu \, I\)) for some \(\mu >0\). If \(r\ge 0\) and \(p>p_0>0\), then

$$\begin{aligned}{} & {} \left( A^\frac{r}{2}B^{p_0}\!A^\frac{r}{2}\right) ^{\frac{m+r}{p_0+r}} \ge ^J \left( A^\frac{r}{2}B^{p}A^\frac{r}{2}\right) ^{\frac{m+r}{p+r}}, \quad \ \ \nonumber \\{} & {} \left( B^\frac{r}{2}A^{p}A^\frac{r}{2}\right) ^{\frac{m+r}{p+r}} \ge ^J \left( B^\frac{r}{2}A^{p_0}\!A^\frac{r}{2}\right) ^{\frac{m+r}{p_0+r}}\ \end{aligned}$$
(10)

where \(m=\min \{p,2p_0+\min \{1,r\}\}\).

The case \(p_0=1\) of Theorem 3.12 implies (EIF). Indeed, if \(p_0=1\), we have \(m=\min \{p,3,2+r\}\) and by Lemma 3.1, after aplying (IL-H) with \(\alpha =\frac{1+r}{m+r}\) to both inequalities in (10), we obtain

$$\begin{aligned} B^{-r}\sharp _{s_1} A^p\ge ^J A \qquad \hbox {and} \qquad B\ge ^J A^{-r}\sharp _{s_1}B^p \end{aligned}$$

for all \(r\ge 0\) and \(p>1\) with \(s_1=\frac{1+r}{p+r}\), whenever \(A, B \in {\mathcal U}^*\). Since \(A\ge ^J B\), this yields Corollary 3.9 and (EIF). As the converse also holds, the essential part of the indefinite Furuta inequality implies the case \(p_0=1\) of Theorem 3.12.

The next corollary is a variant of Theorem 3.12.

Corollary 3.13

Let A, \(B\in {{\mathbb {C}}}^{n\times n}\) be \(J-\)selfadjoint with nonnegative eigenvalues, \(\delta >0\) and \(\mu \, I\ge ^J A^\delta \ge ^J B^\delta \) (or \( A^\delta \ge ^J B^\delta \ge ^J \mu \, I\)) for some \(\mu >0\). If \(r\ge 0\) and \(p>p_0>0\), then (10) holds with \(m=\min \{p,2p_0+\min \{\delta ,r\}\}\).

Finally, we observe that if \(A,B\in \, {\mathcal U}^*\) and \({\textrm{Log}}(A) \ge ^J {\textrm{Log}}(B)\), then (10) holds for \(r\ge 0\) and \(p>p_0>0\) with \(m=\min \{p,2p_0\}\). This corresponds to the indefinite complete form of Furuta inequality for the J-chaotic order [8, Theorem 3.2] and it can be seen as the case \(\delta \rightarrow 0^+\) of Corollary 3.13, recalling that

$$\begin{aligned} {\textrm{Log}}(A)=\lim _{\delta \rightarrow 0^+} \frac{A^\delta -I}{\delta }. \end{aligned}$$

Results in this note can inspire the study of other inequalities of indefinite type for matrices and operators. Application and development of the tools and ideas here discussed is a tempting project, and there is indeed much to antecipate.