Volume 10 (2009), Issue 4, Article 95, 7 pp.
INEQUALITIES FOR J−CONTRACTIONS INVOLVING THE α−POWER MEAN
G. SOARES
CM-UTAD, U NIVERSITY OF T RÁS - OS -M ONTES AND A LTO D OURO
M ATHEMATICS D EPARTMENT
P5000-911 V ILA R EAL
gsoares@utad.pt
Received 22 June, 2009; accepted 14 October, 2009
Communicated by S.S. Dragomir
A BSTRACT. A selfadjoint involutive matrix J endows Cn with an indefinite inner product [·, ·]
given by [x, y] := hJx, yi, x, y ∈ Cn . We present some inequalities of indefinite type involving
the α−power mean and the chaotic order. These results are in the vein of those obtained by E.
Kamei [6, 7].
Key words and phrases: J−selfadjoint matrix, Furuta inequality, J−chaotic order, α−power mean.
2000 Mathematics Subject Classification. 47B50, 47A63, 15A45.
1. I NTRODUCTION
For a selfadjoint involution matrix J, that is, J = J ∗ and J 2 = I, we consider Cn with
the indefinite Krein space structure endowed by the indefinite inner product [x, y] := y ∗ Jx,
x, y ∈ Cn . Let Mn denote the algebra of n × n complex matrices. The J−adjoint matrix A#
of A ∈ Mn is defined by
[A x, y] = [x, A# y], x, y ∈ Cn ,
or equivalently, A# = JA∗ J. A matrix A ∈ Mn is said to be J−selfadjoint if A# = A, that is,
if JA is selfadjoint. For a pair of J−selfadjoint matrices A, B, the J−order relation A ≥J B
means that [Ax, x] ≥ [Bx, x], x ∈ Cn , where this order relation means that the selfadjoint matrix JA − JB is positive semidefinite. If A, B have positive eigenvalues, Log(A) ≥J Log(B)
is called the J−chaotic order, where Log(t) denotes the principal branch of the logarithm function. The J− chaotic order is weaker than the usual J−order relation A ≥J B [11, Corollary
2].
A matrix A ∈ Mn is called a J−contraction if I ≥J A# A. If A is J−selfadjoint and
I ≥J A, then all the eigenvalues of A are real. Furthermore, if A is a J−contraction, by a
theorem of Potapov-Ginzburg [2, Chapter 2, Section 4], all the eigenvalues of the product A# A
are nonnegative.
Sano [11, Corollary 2] obtained the indefinite version of the Löwner-Heinz inequality of
indefinite type, namely for A, B J−selfadjoint matrices with nonnegative eigenvalues such that
I ≥J A ≥J B, then I ≥J Aα ≥J B α , for any 0 ≤ α ≤ 1. The Löwner-Heinz inequality has
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G. S OARES
a famous extension which is the Furuta inequality. An indefinite version of this inequality was
established by Sano [10, Theorem 3.4] and Bebiano et al. [3, Theorem 2.1] in the following
form: Let A, B be J−selfadjoint matrices with nonnegative eigenvalues and µ I ≥J A ≥J B
(or A ≥J B ≥J µ I) for some µ > 0. For each r ≥ 0,
r 1
r 1
r
r
(1.1)
A 2 Ap A 2 q ≥J A 2 B p A 2 q
and
r
r
B 2 Ap B 2
(1.2)
1q
r
r
≥J B 2 B p B 2
1q
hold for all p ≥ 0 and q ≥ 1 with (1 + r)q ≥ p + r.
2. I NEQUALITIES FOR α−P OWER M EAN
For J−selfadjoint matrices A, B with positive eigenvalues, A ≥J B and 0 ≤ α ≤ 1, the
α−power mean of A and B is defined by
1
1
1 α
1
A]α B = A 2 A− 2 BA− 2 A 2 .
1
1
1
1
1
1 α
Since I ≥J A− 2 BA− 2 (or I ≤J A− 2 BA− 2 ) the J−selfadjoint power A− 2 BA− 2
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
and µI ≥J A ≥J B for some µ > 0, then for all p ≥ 1 and r ≥ 0
A−r ] 1+r B p ≤J A
(2.1)
p+r
and
B −r ] 1+r Ap ≥J B.
(2.2)
p+r
The indefinite version of Kamei’s satellite theorem for the Furuta inequality [7] was established in [4] as follows: If A, B are J−selfadjoint matrices with nonnegative eigenvalues and
µI ≥J A ≥J B for some µ > 0, then
A−r ] 1+r B p ≤J B ≤J A ≤J B −r ] 1+r Ap
(2.3)
p+r
p+r
for all p ≥ 1 and r ≥ 0.
Remark 1. Note that by (2.3) and using the fact that X # AX ≥J X # BX for all X ∈ Mn if
r 1+r
r
r 1+r
r
and only if A ≥J B, we have A1+r ≥J A 2 B p A 2 p+r and B 2 Ap B 2 p+r ≥J B 1+r . Applying
1
the Löwner-Heinz inequality of indefinite type, with α = 1+r
, we obtain
r 1
r
r 1
r
A ≥J A 2 B p A 2 p+r and
B 2 Ap B 2 p+r ≥J B
for all p ≥ 1 and r ≥ 0.
In [4], the following extension of Kamei’s satellite theorem of the Furuta inequality was
shown.
Lemma 2.1. Let A, B be J−selfadjoint matrices with nonnegative eigenvalues and µI ≥J
A ≥J B for some µ > 0. Then
A−r ] t+r B p ≤J B t
p+r
and
At ≤J B −r ] t+r Ap ,
p+r
for r ≥ 0 and 0 ≤ t ≤ p.
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 95, 7 pp.
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I NEQUALITIES FOR J−CONTRACTIONS
3
Theorem 2.2. Let A, B be J−selfadjoint matrices with nonnegative eigenvalues and µI ≥J
A ≥J B for some µ > 0. Then
1t
1t
(2.4)
A−r ] 1+r B p ≤J A−r ] t+r B p ≤J B ≤J A ≤J B −r ] t+r Ap ≤J B −r ] 1+r Ap ,
p+r
p+r
p+r
p+r
for r ≥ 0 and 1 ≤ t ≤ p.
Proof. Without loss of generality, we may consider µ = 1, otherwise we can replace A and B
by µ1 A and µ1 B. Let 1 ≤ t ≤ p. Applying the Löwner Heinz inequality of indefinite type in
Lemma 2.1 with α = 1t , we get
1
1
p t
−r
p t
J
J
J
−r
t+r
t+r
.
≤ B≤ A≤ B ] A
A ] B
p+r
p+r
Let A1 = A and B1 = A−r ] t+r B p
1t
. Note that
t
A−r ] 1+r B p = A−r ] 1+r A−r ] t+r B p = A−r
1 ] 1+r B1 .
p+r
(2.5)
p+r
p+r
t+r
t+r
Since µI ≥J A1 ≥J B1 , applying Lemma 2.1 to A1 and B1 , with t = 1 and p = t, we obtain
1t
A−r ] 1+r B p ≤J B1 = A−r ] t+r B p .
p+r
p+r
The remaining inequality in (2.4) can be obtained in an analogous way using the second inequality in Lemma 2.1, with t = 1 and p = t.
Theorem 2.3. Let A, B be J−selfadjoint matrices with nonnegative eigenvalues and µI ≥J
A ≥J B for some µ > 0. Then
t1
t1
t1
t1
A−r ] t1 +r B p 1 ≤J A−r ] t2 +r B p 2 and
B −r ] t1 +r Ap 1 ≥J B −r ] t2 +r Ap 2
p+r
p+r
p+r
p+r
for r ≥ 0 and 1 ≤ t2 ≤ t1 ≤ p.
Proof. Without loss of generality, we may consider µ = 1, otherwise we can replace A and B
1
1
1
−r
p t2
by µ A and µ B. Let A1 = A and B1 = A ] t2 +r B
. By Lemma 2.1 and the Löwner Heinz
p+r
inequality of indefinite type with α = t12 , we have B1 ≤J B ≤J A1 ≤J I. Applying Lemma
2.1 to A1 and B1 , with p = t2 , we obtain
t1
t2
t1
−r
J
−r
p t2
.
(2.6)
A1 ] t1 +r B1 ≤ B1 = A ] t2 +r B
t2 +r
p+r
On the other hand,
(2.7)
p
A−r
1 ] t1 +r B
p+r
=
A−r
1 ] t1 +r
−r
A ] t2 +r B
t2 +r
p
t1 t2
2
t2
= A−r
1 ] t1 +r B1 .
t2 +r
p+r
By (2.6) and (2.7),
−r
p
J
A ] t1 +r B ≤
p+r
−r
A ] t2 +r B
p
tt1
2
.
p+r
Using the Löwner-Heinz inequality of indefinite type with α = t11 , we have
t1
t1
A−r ] t1 +r B p 1 ≤J A−r ] t2 +r B p 2 .
p+r
The remaining inequality can be obtained analogously.
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 95, 7 pp.
p+r
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4
G. S OARES
Theorem 2.4. Let A, B be J−selfadjoint matrices with nonnegative eigenvalues and µI ≥J
A ≥J B for some µ > 0. Then
t
t
A−r ] t+r B p ≤J A−r ] 1+r B p ≤J B t ≤J At ≤J B −r ] 1+r Ap ≤J B −r ] t+r Ap
p+r
p+r
p+r
p+r
for 0 ≤ t ≤ 1 ≤ p and r ≥ 0.
Proof. By the indefinite version of Kamei’s satellite theorem for the Furuta inequality and since
0 ≤ t ≤ 1, we can apply the Löwner-Heinz inequality of indefinite type with α = t, to get
t
t
A−r ] 1+r B p ≤J B t ≤J At ≤J B −r ] 1+r Ap .
p+r
p+r
Note that
−r
p
A ] t+r B = A
p+r
r
t −t
] t+r
1+r
Since µI ≥J At , for all t > 0 [10] and At ≥J
−r
A ] 1+r B
p
p+r
A−r ] 1+r B p
p+r
t
t 1t
.
, applying the indefinite ver-
t
sion
of Kamei’s
t satellite theorem for the Furuta inequality with A and B replaced by A and
A−r ] 1+r B p , respectively, and with r replaced by r/t and p replaced by 1/t, we have
p+r
t
A−r ] t+r B p ≤J A−r ] 1+r B p .
p+r
p+r
The remaining inequality can be obtained analogously.
3. I NEQUALITIES I NVOLVING THE J−C HAOTIC O RDER
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);
r
r r
(ii) A 2 B p A 2 p+r ≤J Ar , for all p ≥ 0 and r ≥ 0;
r r
r
(iii) B 2 Ap B 2 p+r ≥J B r , for all p ≥ 0 and r ≥ 0.
Under the chaotic order Log (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]). If A, B are J−selfadjoint matrices with positive eigenvalues and A ≥J B,
then B −1 ≥J A−1 .
Lemma 3.3 ([10]). Let A, B be J−selfadjoint matrices with positive eigenvalues and I ≥J A,
I ≥J B. Then
1
λ−1 1
1
λ
2 12
2
2
(ABA) = AB B A B
B 2 A,
λ ∈ R.
Theorem 3.4 (Satellite theorem of the chaotic Furuta inequality). Let A, B be J−selfadjoint
matrices with positive eigenvalues and µI ≥J A, µI ≥J B for some µ > 0. If Log (A) ≥J
Log (B) then
A−r ] 1+r B p ≤J B and B −r ] 1+r Ap ≥J A
p+r
p+r
for all p ≥ 1 and r ≥ 0.
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 95, 7 pp.
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I NEQUALITIES FOR J−CONTRACTIONS
5
Proof. Let Log (A) ≥J Log (B). Interchanging the roles of r and p in Theorem 3.1 from the
equivalence between (i) and (iii), we obtain
p
p+r
p
p
r
(3.1)
B2A B2
≥J B p ,
for all p ≥ 0 and r ≥ 0. From Lemma 3.3, we get
p p−1
p
p
p
p
p − p+r
r
r 1+r
r
p
−
r
− r2
p+r
B2.
A
A2 B A2
A 2 = B2 B2A B2
Hence, applying Lemma 3.2 to (3.1), noting that 0 ≤ (p−1)/p ≤ 1 and using the Löwner-Heinz
inequality of indefinite type, we have
p
p
r
r
r 1+r
r
A− 2 A 2 B p A 2 p+r A− 2 ≤J B 2 B 1−p B 2 = B.
The result now follows easily. The remaining inequality can be analogously obtained.
As a generalization of Theorem 3.4, we can obtain the next characterization of the chaotic
order.
Theorem 3.5. 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 equivalent:
(i) Log (A) ≥J Log (B);
(ii) A−r ] t+r B p ≤J B t , for r ≥ 0 and 0 ≤ t ≤ p;
p+r
(iii) B −r ] t+r Ap ≥J At , for r ≥ 0 and 0 ≤ t ≤ p;
p+r
(iv) A−r ] −t+r B p ≤J A−t , for r ≥ 0 and 0 ≤ t ≤ r;
p+r
(v) B −r ] −t+r Ap ≥J B −t , for r ≥ 0 and 0 ≤ δ ≤ r.
p+r
Proof. We first prove the equivalence between (i) and (iv). By Theorem 3.1, Log(A) ≥J
r
r r
Log(B) is equivalent to A 2 B p A 2 p+r ≤J Ar , for all p ≥ 0 and r ≥ 0. Henceforth, since
0 ≤ t ≤ r applying the Löwner-Heinz inequality of indefinite type, we easily obtain
−t+r
r i
−t+r h r p r p+r
r
r
p r2 p+r
2
2
2
A B A
= A B A
≤J Ar−t .
Analogously, using the equivalence between (i) and (iii) in Theorem 3.1, we easily obtain that
(i) is equivalent to (v).
(ii)⇔ (v) Suppose that (ii) holds. By Lemma 3.3 and using the fact that X # AX ≥J X # BX
for all X ∈ Mn if and only if A ≥J B, we have
r
r
r t+r
r
A 2 B t A 2 ≥J A 2 B p A 2 p+r
t−p
p
p+r
p
p
p
r
r
r
B 2 A2 .
= A2 B 2 B 2 A B 2
It easily follows by Lemma 3.2, that
−t+p
p
p
p+r
,
B p−t ≤J B 2 Ar B 2
for r ≥ 0 and 0 ≤ t ≤ p. Replacing p by r, we obtain (v).
In an analogous way, we can prove that (v)⇔(iii).
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 95, 7 pp.
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6
G. S OARES
Remark 2. Consider two J−selfadjoint matrices A, B with positive eigenvalues and µI ≥J A,
µI ≥J B for some µ > 0. Let 1 ≤ t ≤ p. Applying the Löwner Heinz inequality of indefinite
type in Theorem 3.5 (ii) with α = 1t , we obtain that Log (A) ≥J Log (B) if and only if
1t
A−r ] t+r B p ≤J B.
p+r
1t
Consider A1 = A and B1 = A−r ] t+r B p . Following analogous steps to the proof of Theop+r
rem 2.2 we have
t
A−r ] 1+r B p = A−r
1 ] 1+r B1 .
p+r
J
J
t+r
J
Since B1 ≤ B ≤ µI and A1 ≤ µI, applying Theorem 3.5 (ii) to A1 and B1 , with t = 1 and
p = t, we obtain Log (A1 ) ≥J Log (B1 ) if and only if
1
−r
p
J
−r
p t
A ] 1+r B ≤ A ] t+r B
.
p+r
p+r
Note that Log (A1 ) ≥J Log (B1 ) is equivalent to Log (A) ≥J Log (B), when r −→ 0+ . In
this way we can easily obtain Corollary 3.6, Corollary 3.8 and Corollary 3.8 from Theorem 3.5:
Corollary 3.6. Let A, B be J−selfadjoint matrices with positive eigenvalues and µI ≥J A,
µI ≥J B for some µ > 0. Then Log (A) ≥J Log (B) if and only if
1t
1t
A−r ] 1+r B p ≤J A−r ] t+r B p ≤J B and A ≤J B −r ] t+r Ap ≤J B −r ] 1+r Ap ,
p+r
p+r
p+r
p+r
for r ≥ 0 and 1 ≤ t ≤ p.
Corollary 3.7. Let A, B be J−selfadjoint matrices with positive eigenvalues and µI ≥J A,
µI ≥J B for some µ > 0. Then Log (A) ≥J Log (B) if and only if
1
1
1
1
−r
p t1
J
−r
p t2
−r
p t1
J
−r
p t2
A ] t1 +r B
≤ A ] t2 +r B
and
B ] t1 +r A
≥ B ] t2 +r A
p+r
p+r
p+r
p+r
for r ≥ 0 and 1 ≤ t2 ≤ t1 ≤ p.
Corollary 3.8. Let A, B be J−selfadjoint matrices with positive eigenvalues and µI ≥J A,
µI ≥J B for some µ > 0. Then Log (A) ≥J Log (B) if and only if
t
t
A−r ] t+r B p ≤J A−r ] 1+r B p ≤J B t and At ≤J B −r ] 1+r Ap ≤J B −r ] t+r Ap ,
p+r
p+r
p+r
p+r
for r ≥ 0 and 0 ≤ t ≤ 1 ≤ p.
R EFERENCES
[1] T. ANDO, Löwner inequality of indefinite type, Linear Algebra Appl., 385 (2004), 73–84.
[2] T.Ya. AZIZOV AND I.S. IOKHVIDOV, Linear Operators in Spaces with an Indefinite Metric,
Nauka, Moscow, 1986, English Translation: Wiley, New York, 1989.
[3] N. BEBIANO, R. LEMOS, J. da PROVIDÊNCIA
Furuta inequality of indefinite type, preprint.
AND
G. SOARES, Further developments of
[4] N. BEBIANO, R. LEMOS, J. da PROVIDÊNCIA
J−contractions, preprint.
AND
G. SOARES, Operator inequalities for
[5] M. FUJII, T. FURUTA AND E. 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.
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 95, 7 pp.
http://jipam.vu.edu.au/
I NEQUALITIES FOR J−CONTRACTIONS
7
[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. JIANG AND E. KAMEI, Characterization of chaotic order and its application to
Furuta inequality, Proc. Amer. Math. Soc., 125 (1997), 3655–3658.
[10] T. SANO, Furuta inequality of indefinite type, Math. Inequal. Appl., 10 (2007), 381–387.
[11] T. SANO, On chaotic order of indefinite type, J. Inequal. Pure Appl. Math., 8(3) (2007), Art. 62.
[ONLINE: http://jipam.vu.edu.au/article.php?sid=890]
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 95, 7 pp.
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