INEQUALITIES FOR J−CONTRACTIONS
INVOLVING THE α−POWER MEAN
Inequalities for J−contractions
G. Soares
G. SOARES
CM-UTAD, University of Trás-os-Montes and Alto Douro
Mathematics Department
P5000-911 Vila Real
EMail: gsoares@utad.pt
vol. 10, iss. 4, art. 95, 2009
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Contents
Received:
22 June, 2009
Accepted:
14 October, 2009
Communicated by:
JJ
II
S.S. Dragomir
J
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2000 AMS Sub. Class.:
47B50, 47A63, 15A45.
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Key words:
J−selfadjoint matrix, Furuta inequality, J−chaotic order, α−power mean.
Abstract:
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n
A selfadjoint involutive matrix J endows C 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].
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Contents
1
Introduction
3
2
Inequalities for α−Power Mean
5
3
Inequalities Involving the J−Chaotic Order
10
Inequalities for J−contractions
G. Soares
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1.
Introduction
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 ,
Inequalities for J−contractions
G. Soares
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 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
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some µ > 0. For each r ≥ 0,
(1.1)
r
r
1q
≥J A 2 B p A 2
r
r
1q
≥J B 2 B p B 2
A 2 Ap A 2
r
r
1q
r
r
1q
and
(1.2)
B 2 Ap B 2
Inequalities for J−contractions
hold for all p ≥ 0 and q ≥ 1 with (1 + r)q ≥ p + r.
G. Soares
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2.
Inequalities for α−Power Mean
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
1 α
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
−r
p
J
A ] 1+r B ≤
(2.1)
p+r
A
and
Inequalities for J−contractions
G. Soares
vol. 10, iss. 4, art. 95, 2009
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B −r ] 1+r Ap ≥J B.
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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
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(2.2)
(2.3)
p+r
A−r ] 1+r B p ≤J B ≤J A ≤J B −r ] 1+r Ap
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 ∈
r
r 1+r
r
r 1+r
Mn if 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
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B 1+r . Applying the Löwner-Heinz inequality of indefinite type, with α =
obtain
r
r 1
r
r 1
A ≥J A 2 B p A 2 p+r and
B 2 Ap B 2 p+r ≥J B
1
,
1+r
we
for all p ≥ 1 and r ≥ 0.
In [4], the following extension of Kamei’s satellite theorem of the Furuta inequality was shown.
Inequalities for J−contractions
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
G. Soares
vol. 10, iss. 4, art. 95, 2009
At ≤J B −r ] t+r Ap ,
p+r
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for r ≥ 0 and 0 ≤ t ≤ p.
Contents
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
A−r ] 1+r B p ≤J A−r ] t+r B p
(2.4)
p+r
p+r
1
J
J
J
−r
p t
≤ B ≤ A ≤ B ] t+r A
≤J B −r ] 1+r Ap ,
p+r
A ] t+r B
p+r
J
I
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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
−r
II
p+r
for r ≥ 0 and 1 ≤ t ≤ p.
JJ
p
1t
J
J
J
≤ B≤ A≤
−r
B ] t+r A
p+r
p
1t
.
Close
−r
Let A1 = A and B1 = A ] t+r B
p
1t
. Note that
−r
p
−r
−r
p
t
A ] 1+r B = A ] 1+r A ] t+r B = 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 .
G. Soares
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
1
1
1
1
J
−r
p t2
−r
p t1
J
−r
p t2
−r
p t1
t
+r
t
+r
t
+r
t
+r
≤ A ]2 B
and
B ]1 A
≥ B ]2 A
A ]1 B
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
1
1
1
−r
p t2
A and B by µ A and µ B. Let A1 = A and B1 = A ] t2 +r B
. By Lemma 2.1
p+r
and the Löwner Heinz 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
t
+r
t
+r
(2.6)
A1 ] 1 B1 ≤ B1 = A ] 2 B
.
t2 +r
p+r
On the other hand,
(2.7)
p
A−r
1 ] t1 +r B
p+r
=
Inequalities for J−contractions
p+r
p+r
A−r
1 ] t1 +r
t2 +r
−r
A ] t2 +r B
p+r
p
t1 t2
2
t2
= A−r
1 ] t1 +r B1 .
t2 +r
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By (2.6) and (2.7),
−r
p
J
A ] t1 +r B ≤
−r
A ] t2 +r B
p+r
p
tt1
2
.
p+r
Using the Löwner-Heinz inequality of indefinite type with α =
−r
A ] t1 +r B
p
t1
1
J
≤
−r
A ] t2 +r B
p+r
p
t1
2
1
,
t1
we have
.
p+r
Inequalities for J−contractions
G. Soares
The remaining inequality can be obtained analogously.
vol. 10, iss. 4, art. 95, 2009
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
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p+r
p+r
for 0 ≤ t ≤ 1 ≤ p and r ≥ 0.
JJ
II
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 .
J
I
p+r
p+r
Note that
A−r ] t+r B p = A
p+r
r
t −t
] t+r
1+r
A−r ] 1+r B p
p+r
t 1t
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.
t
Since µI ≥J At , for all t > 0 [10] and At ≥J A−r ] 1+r B p , applying the inp+r
definite version of Kamei’s satellite theorem for the Furuta inequality with A and B
t
replaced by At and A−r ] 1+r B p , respectively, and with r replaced by r/t and p
p+r
replaced by 1/t, we have
t
−r
p
J
−r
p
A ] t+r B ≤ A ] 1+r B
.
p+r
p+r
The remaining inequality can be obtained analogously.
Inequalities for J−contractions
G. Soares
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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:
J
(i) Log(A) ≥ 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.
J
Under the chaotic order Log (A) ≥ 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
1
1 λ−1
1
(ABA)λ = AB 2 B 2 A2 B 2
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
p+r
and
B −r ] 1+r Ap ≥J A
p+r
Inequalities for J−contractions
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vol. 10, iss. 4, art. 95, 2009
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for all p ≥ 1 and r ≥ 0.
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 p2 p+r
2
(3.1)
B AB
≥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
r 1+r
−
r
− r2
p
p+r
B2.
A 2 = B2 B2A B2
A
A2 B A2
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
A
− r2
r
2
A BpA
r
2
1+r
p+r
A
− r2
p
2
p
2
≤J B B 1−p B = 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
Inequalities for J−contractions
G. Soares
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(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
h r
i −t+r
r r
r
r −t+r
r
≤J Ar−t .
A 2 B p A 2 p+r = A 2 B p A 2 p+r
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
A 2 B t A 2 ≥J
=
r
r t+r
A 2 B p A 2 p+r
p
t−p p r
p
r
r p2 p+r
2
2
2
A B B AB
B 2 A2 .
It easily follows by Lemma 3.2, that
B
p−t
J
≤
Inequalities for J−contractions
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p
2
r
B AB
p
2
−t+p
p+r
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for r ≥ 0 and 0 ≤ t ≤ p. Replacing p by r, we obtain (v).
In an analogous way, we can prove that (v)⇔(iii).
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
Consider A1 = A and B1 =
proof of Theorem 2.2 we have
A−r ] t+r B p
p+r
1t
. Following analogous steps to the
Inequalities for J−contractions
G. Soares
t
A−r ] 1+r B p = A−r
1 ] 1+r B1 .
p+r
vol. 10, iss. 4, art. 95, 2009
t+r
Since B1 ≤J B ≤J µI and A1 ≤J 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
1t
A−r ] 1+r B p ≤J A−r ] t+r B p .
p+r
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
1
1
−r
p
J
−r
p t
J
J
−r
p t
t+r
t+r
1+r
A ] B ≤ A ] B
≤ B and A ≤ B ] A
≤J B −r ] 1+r Ap ,
p+r
p+r
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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
J
−r
−r
J
−r
p t1
p t2
p t1
p t2
t
+r
t
+r
t
+r
t
+r
≤ A ]2 B
and
B ]1 A
≥ B ]2 A
A ]1 B
p+r
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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:
p+r
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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
for r ≥ 0 and 0 ≤ t ≤ 1 ≤ p.
p+r
p+r
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References
[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 AND G. SOARES, Further
developments of Furuta inequality of indefinite type, preprint.
[4] N. BEBIANO, R. LEMOS, J. da PROVIDÊNCIA AND G. SOARES, Operator
inequalities for J−contractions, preprint.
[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.
[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.
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[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]
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