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Inequalities forJ₋contractions

G. Soares

vol. 10, iss. 4, art. 95, 2009

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INEQUALITIES FOR

J

−

CONTRACTIONS

INVOLVING THE

α

−

POWER MEAN

G. SOARES

CM-UTAD, University of Trás-os-Montes and Alto Douro Mathematics Department

P5000-911 Vila Real EMail:gsoares@utad.pt

Received: 22 June, 2009

Accepted: 14 October, 2009

Communicated by: S.S. Dragomir

2000 AMS Sub. Class.: 47B50, 47A63, 15A45.

Key words: J−selfadjoint matrix, Furuta inequality,J−chaotic order,α−power mean.

Abstract: A selfadjoint involutive matrixJendowsCn

with an indefinite inner product[·,·] given by[x, y] :=hJ x, yi,x, y∈Cn

Inequalities forJ

−contractions

G. Soares

vol. 10, iss. 4, art. 95, 2009

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Contents

1 Introduction 3

2 Inequalities forα−Power Mean 5

Inequalities forJ

−contractions

G. Soares

vol. 10, iss. 4, art. 95, 2009

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1.

Introduction

For a selfadjoint involution matrixJ, that is,J = J∗ _{and J}2 _{= I,we considerC}n

with the indefinite Krein space structure endowed by the indefinite inner product

[x, y] := y∗_{Jx, x, y ∈ C}n_{. LetM}

n denote the algebra ofn×n complex matrices.

TheJ−adjoint matrixA#_ofA∈M

nis defined by

[A x, y] = [x, A#y], x, y ∈Cn_,

or equivalently, A# _{= JA}∗_{J. A matrix A ∈ M}

n is said to be J−selfadjoint if

A#_{=A, that is, ifJAis selfadjoint. For a pair ofJ−selfadjoint matricesA, B, the}

J−order relationA ≥J B means that[Ax, x] ≥ [Bx, x], x ∈ Cn_{, where this order}

relation means that the selfadjoint matrixJA−JBis positive semidefinite. IfA,B

have positive eigenvalues,Log(A)≥J Log(B)is called theJ−chaotic order, where Log(t)denotes the principal branch of the logarithm function. TheJ−chaotic order is weaker than the usualJ−order relationA≥J B [11, Corollary 2].

A matrixA∈Mnis called aJ−contraction ifI ≥J A#A. IfAisJ−selfadjoint

andI ≥J A, then all the eigenvalues ofAare real. Furthermore, ifAis aJ−contraction, by a theorem of Potapov-Ginzburg [2, Chapter 2, Section 4], all the eigenvalues of the productA#_{Aare nonnegative.}

Sano [11, Corollary 2] obtained the indefinite version of the Löwner-Heinz in-equality of indefinite type, namely forA, B J−selfadjoint matrices with nonnegative eigenvalues such thatI ≥J A≥J B, thenI ≥J Aα _≥J _Bα_{, for any0≤α≤1.The}

Inequalities forJ

−contractions

G. Soares

vol. 10, iss. 4, art. 95, 2009

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someµ >0. For eachr≥0,

(1.1) Ar2ApA

r

2 1

q _≥J _Ar_2Bp_Ar₂

1

q

and

(1.2) Br2ApB

r

2 1

q

≥J Br2BpB

r

2 1

q

Inequalities forJ

2.

Inequalities for

α

−

Power Mean

ForJ−selfadjoint matricesA, Bwith positive eigenvalues,A ≥J B and0≤α≤1,

2) theJ−selfadjoint power

A−12BA− 1 2

α

is well defined.

The essential part of the Furuta inequality of indefinite type can be reformulated in terms of α−power means as follows. If A, B are J−selfadjoint matrices with nonnegative eigenvalues andµI ≥J A≥J B for someµ >0, then for allp≥1and

Inequalities forJ

B1+r_{. Applying the Löwner-Heinz inequality of indefinite type, with α =} 1 1+r, we

In [4], the following extension of Kamei’s satellite theorem of the Furuta inequal-ity was shown.

Lemma 2.1. LetA, B beJ−selfadjoint matrices with nonnegative eigenvalues and µI ≥J A≥J B for someµ >0. Then

Theorem 2.2. LetA, BbeJ−selfadjoint matrices with nonnegative eigenvalues and µI ≥J A≥J B for someµ >0. Then

Proof. Without loss of generality, we may considerµ= 1, otherwise we can replace

AandB by _µ1Aand _µ1B. Let1≤ t ≤ p. Applying the Löwner Heinz inequality of indefinite type in Lemma2.1withα = 1

Inequalities forJ

The remaining inequality in (2.4) can be obtained in an analogous way using the second inequality in Lemma2.1, witht= 1andp=t.

Theorem 2.3. LetA, BbeJ−selfadjoint matrices with nonnegative eigenvalues and µI ≥J A≥J B for someµ >0. Then

Proof. Without loss of generality, we may considerµ= 1, otherwise we can replace

AandB by 1_µAand _µ1B. LetA1 = AandB1 =

and the Löwner Heinz inequality of indefinite type with α = 1

t2,we have B1 ≤

On the other hand,

Inequalities forJ

Using the Löwner-Heinz inequality of indefinite type withα= 1

t1,we have

The remaining inequality can be obtained analogously.

Theorem 2.4. LetA, BbeJ−selfadjoint matrices with nonnegative eigenvalues and µI ≥J A≥J B for someµ >0. Then

Proof. By the indefinite version of Kamei’s satellite theorem for the Furuta

inequal-ity and since0 ≤ t ≤ 1, we can apply the Löwner-Heinz inequality of indefinite type withα=t, to get

Inequalities forJ

−contractions

G. Soares

vol. 10, iss. 4, art. 95, 2009

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replaced byAt _andA−r_♯1+r p+rB

pt_{,respectively, and withr replaced byr/tandp}

replaced by1/t, we have

A−r_♯t+r p+rB

p _≤J _A−r_♯1+r p+rB

pt_.

Inequalities forJ

−contractions

G. Soares

vol. 10, iss. 4, art. 95, 2009

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3.

Inequalities Involving the

J

−

Chaotic Order

The following theorem is the indefinite version of the Chaotic Furuta inequality, a result previously stated in the context of Hilbert spaces by Fujii, Furuta and Kamei [5].

Theorem 3.1. Let A, B be J−selfadjoint matrices with positive eigenvalues and µI ≥J A, µI ≥J B for some µ >0. Then the following statements are mutually equivalent:

(i) Log(A)≥J Log(B);

(ii) Ar2BpA

r

2

r

p+r _≤J _Ar_{, for allp≥0andr≥0;}

(iii) Br2ApB

r

2

r

p+r _≥J _Br_{, for allp≥0andr≥0.}

Under the chaotic orderLog (A)≥J Log (B), we can obtain the satellite theorem of the Furuta inequality. To prove this result, we need the following lemmas.

Lemma 3.2 ([10]). IfA, BareJ−selfadjoint matrices with positive eigenvalues and A≥J B, thenB−1 _≥J _A−1_.

Lemma 3.3 ([10]). Let A, B be J−selfadjoint matrices with positive eigenvalues andI ≥J A, I ≥J B.Then

(ABA)λ =AB12

B12A2B 1 2

λ−1

B12A, λ _∈R.

Theorem 3.4 (Satellite theorem of the chaotic Furuta inequality). LetA, B be J−selfadjoint matrices with positive eigenvalues andµI ≥J A,µI ≥J B for some µ >0. IfLog (A)≥J Log (B)then

A−r_♯1+r p+rB

p _≤J _{B and B}−r_♯1+r p+rA

Inequalities forJ

from the equivalence between (i) and (iii), we obtain

(3.1) Bp2ArB Löwner-Heinz inequality of indefinite type, we have

A−r2 A

The result now follows easily. The remaining inequality can be analogously ob-tained.

As a generalization of Theorem 3.4, we can obtain the next characterization of the chaotic order.

Inequalities forJ

Proof. We first prove the equivalence between (i) and (iv). By Theorem3.1,Log(A)≥J

Log(B)is equivalent to Ar2BpA

r

2

r

p+r _≤J _Ar_{, for allp≥0andr≥0. Henceforth,}

since0 ≤ t ≤rapplying the Löwner-Heinz inequality of indefinite type, we easily obtain

Analogously, using the equivalence between (i) and (iii) in Theorem3.1, we easily obtain that (i) is equivalent to (v).

(ii)⇔(v) Suppose that (ii) holds. By Lemma3.3and using the fact thatX#_{AX ≥}J

It easily follows by Lemma3.2, that

Bp−t_≤J _Bp_2Ar_Bp₂

−t+p

p+r

,

forr ≥0and0≤t≤p. Replacingpbyr, we obtain (v). In an analogous way, we can prove that (v)⇔(iii).

Remark 2. Consider twoJ−selfadjoint matricesA, B with positive eigenvalues and

Inequalities forJ Heinz inequality of indefinite type in Theorem3.5 (ii) with α = 1

t, we obtain that

. Following analogous steps to the proof of Theorem2.2we have

A−r_♯

r−→0+_{.In this way we can easily obtain Corollary3.6, Corollary3.8and Corollary}

3.8from Theorem3.5:

Corollary 3.6. Let A, B be J−selfadjoint matrices with positive eigenvalues and µI ≥J A,µI ≥J B for someµ >0. ThenLog (A)≥J Log (B)if and only if

Inequalities forJ

−contractions

G. Soares

vol. 10, iss. 4, art. 95, 2009

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forr ≥0and1≤t2 ≤t1 ≤p.

Corollary 3.8. Let A, B be J−selfadjoint matrices with positive eigenvalues and µI ≥J A,µI ≥J B for someµ >0. ThenLog (A)≥J Log (B)if and only if

A−r_♯t+r p+rB

p _≤J _A−r_♯1+r p+rB

pt_≤J _Bt

and At_≤J _B−r_♯1+r p+rA

pt_≤J _B−r_♯t+r p+rA

p_,

Inequalities forJ

−contractions

G. Soares

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References

[1] T. ANDO, Löwner inequality of indefinite type, Linear Algebra Appl., 385 (2004), 73–84.

[2] T.Ya. AZIZOVANDI.S. IOKHVIDOV, Linear Operators in Spaces with an

In-definite Metric, Nauka, Moscow, 1986, English Translation: Wiley, New York,

1989.

[3] N. BEBIANO, R. LEMOS, J. da PROVIDÊNCIA AND G. SOARES, Further developments of Furuta inequality of indefinite type, preprint.

[4] N. BEBIANO, R. LEMOS, J. da PROVIDÊNCIAANDG. SOARES, Operator

inequalities forJ−contractions, preprint.

[5] M. FUJII, T. FURUTAANDE. KAMEI, Furuta’s inequality and its application

to Ando’s theorem, Linear Algebra Appl., 179 (1993), 161–169.

[6] E. KAMEI, Chaotic order and Furuta inequality, Scientiae Mathematicae

Japonicae, 53(2) (2001), 289–293.

[7] E. KAMEI, A satellite to Furuta’s inequality, Math. Japon., 33 (1988), 883– 886.

[8] E. KAMEI, Parametrization of the Furuta inequality, Math. Japon., 49 (1999), 65–71.

[9] M. FUJII, J.-F. JIANGANDE. KAMEI, Characterization of chaotic order and

its application to Furuta inequality, Proc. Amer. Math. Soc., 125 (1997), 3655– 3658.

Inequalities forJ

−contractions

G. Soares

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[11] T. SANO, On chaotic order of indefinite type, J. Inequal. Pure Appl. Math.,