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 J2 = I,we considerCn
with the indefinite Krein space structure endowed by the indefinite inner product
[x, y] := y∗Jx, x, y ∈ Cn. 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 Ar2BpAr2
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 1µAand 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 Bp2ArBp2
−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
vol. 10, iss. 4, art. 95, 2009
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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
vol. 10, iss. 4, art. 95, 2009
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[11] T. SANO, On chaotic order of indefinite type, J. Inequal. Pure Appl. Math.,
