ON CHAOTIC ORDER OF INDEFINITE TYPE

(1)

Chaotic Order of Indefinite Type

Takashi Sano vol. 8, iss. 3, art. 62, 2007

Title Page

Contents

JJ II

J I

Page1of 7 Go Back Full Screen

Close

ON CHAOTIC ORDER OF INDEFINITE TYPE

TAKASHI SANO

Department of Mathematical Sciences Faculty of Science, Yamagata University Yamagata 990-8560, Japan

EMail:sano@sci.kj.yamagata-u.ac.jp

Received: 02 June, 2006

Accepted: 27 April, 2007

Communicated by: F. Hansen 2000 AMS Sub. Class.: 47B50, 47A63.

Key words: Inner product space; Furuta inequality of indefinite type.

Abstract: LetA, BbeJ-selfadjoint matrices with positive eigenvalues andI=^J A, I=^J B.Then it is proved as an application of Furuta inequality of indefinite type that

LogA=^J LogB

if and only if

A^r =^J (A^r²B^pA^r²)^p+r^r for allp >0andr >0.

(2)

Title Page Contents

JJ II

J I

Close

In [2], T. Ando gave inequalities for matrices on an (indefinite) inner product space; for instance,

Proposition 1 ([2, Theorem 4]). LetA, BbeJ-selfadjoint matrices withσ(A), σ(B)j (α, β).Then

A=^J B ⇒f(A)=^J f(B) for any operator monotone functionf(t)on(α, β).

Since the principal branch Log x of the logarithm is operator monotone, as a corollary, we have

Corollary 2. For J-selfadjoint matrices A, B with positive eigenvalues and A =^J B, we have

LogA=^J LogB.

In this note, we give a characterization of this inequality relation, called a chaotic order, forJ-selfadjoint matricesA, B with positive eigenvalues andI =^J A, I =^J B.

Before giving our theorem, we recall basic facts about matrices on an (indefinite) inner product space. We refer the reader to [3].

LetM_n(C)be the set of all complexn-square matrices acting onCⁿand leth·,·i be the standard inner product onCⁿ;hx, yi := Pn

i=1xiyi forx = (xi), y = (yi) ∈ Cⁿ. For a selfadjoint involutionJ ∈ Mn(C);J = J^∗ andJ² = I,we consider the (indefinite) inner product[·,·]onCⁿgiven by

[x, y] :=hJ x, yi (x, y ∈Cⁿ).

(3)

Title Page Contents

JJ II

J I

Page3of 7 Go Back Full Screen

Close

TheJ-adjoint matrixA^]ofA∈M_n(C)is defined as [Ax, y] = [x, A^]y] (x, y ∈Cⁿ).

In other words, A^] = J A^∗J. A matrix A ∈ M_n(C) is said to be J-selfadjoint if A^] =AorJ A^∗J =A.And forJ-selfadjoint matricesAandB,theJ-order, denoted asA=^J B, is defined by

[Ax, x]=[Bx, x] (x∈Cⁿ).

A matrixA∈M_n(C)is calledJ-positive ifA=^J O,or [Ax, x]=0 (x∈Cⁿ).

A matrixA∈Mn(C)is said to be aJ-contraction ifI =^J A^]Aor[x, x]=[Ax, Ax]

(x∈Cⁿ).We remark thatI =^J Aimplies that all eigenvalues ofAare real. Hence, for aJ-contractionAall eigenvalues ofA^]Aare real. In fact, by a result of Potapov- Ginzburg (see [3, Chapter 2, Section 4]), all eigenvalues ofA^]Aare non-negative.

We also recall facts in [6]:

Proposition 3 ([6, Theorem 2.6]). Let A, B be J-selfadjoint matrices with non- negative eigenvalues and0< α <1. If

I =^J A=^J B, thenJ-selfadjoint powersA^α, B^α are well defined and

I =^J A^α =^J B^α.

Proposition 4 ([6, Lemma 3.1]). Let A, B be J-selfadjoint matrices with non- negative eigenvalues and I =^J A, I =^J B. Then the eigenvalues of ABA are non-negative and

I =^J A^λ for allλ >0.

(4)

Title Page Contents

JJ II

J I

Page4of 7 Go Back Full Screen

Close

We also have a generalization; Furuta inequality of indefinite type:

Proposition 5 ([6, Theorem 3.4]). Let A, B be J-selfadjoint matrices with non- negative eigenvalues andI =^J A=^J B.For eachr=0,

A^r²A^pA^r²¹_q

=^J A^r²B^pA^r²¹_q holds for allp=0, q =1with(1 +r)q=p+r.

Remark 1. Let0< α <1.ForJ-selfadjoint matricesA, Bwith positive eigenvalues andA=^J B, we have

A^α =^J B^α,

by applying Proposition 1 to the operator monotone function x^α whose principal branch is considered. Hence,

A^α−I

α =^J B^α−I α .

We remark thatA^αis given by the Dunford integral and that A^α−I

α = 1

2πi Z

C

ζ^α−1

α (ζI−A)⁻¹dζ,

whereC is a closed rectifiable contour in the domain of ζ^α with positive direction surrounding all eigenvalues ofAin its interior. Since

ζ^α−1

α →Log ζ (α→0) uniformly forζ,we also have Corollary2.

Our theorem is as follows:

(5)

Title Page Contents

JJ II

J I

Close

Theorem 6. LetA, B beJ-selfadjoint matrices with positive eigenvalues andI =^J A, I =^J B.Then the following statements are equivalent:

(i) LogA =^J LogB.

(ii) A^r =^J (A^r²B^pA^r²)^p+r^r for allp > 0andr >0.

Here, principal branches of the functions are considered.

This theorem, as well as the corresponding result on a Hilbert space ([1,4,5,7]), can be obtained and the similar approach in [7] also works. But careful arguments are necessary, and this is the reason for the present note.

Proof. (ii)=⇒(i): Assume that

A^r =^J A^r²B^pA^r²_p+r^r for allp >0andr >0.Then by Corollary2, we have

r(p+r)LogA=^J rLog A^r²B^pA^r² .

Dividing this inequality byr >0and takingp, rasp= 1, r→0,we have (i).

(i)=⇒(ii): Since

I =^J A, B, by assumption, it follows from Corollary2that

O= LogI =^J LogA,LogB.

Hence, forn∈N

I =^J I + 1

n LogA=:A₁, I+ 1

n LogB =:B₁.

(6)

Title Page Contents

JJ II

J I

Close

For a sufficiently largen, all eigenvalues ofA₁, B₁ are positive. Applying Proposi- tion5toA₁, B₁andnp, nr,^nr+np_nr (resp.) asp, r, q(resp.), we get

(]) A^nr₁ =^J

A

nr 2

1 B₁^npA

nr 2

1

_np+nr^nr

for allp >0, q >0.Recall that

n→∞lim

I+ A n

n

=e^A

for any matrixAand thate^Log^X =Xfor any matrixXwith all eigenvalues positive.

Therefore, takingnasn → ∞in the inequality (]), we obtain the conclusion.

(7)

Title Page Contents

JJ II

J I

Close

References

[1] T. ANDO, On some operator inequalities, Math. Ann., 279 (1987), 157–159.

[2] T. ANDO, Löwner inequality of indefinite type, Linear Algebra Appl., 385 (2004), 73–80.

[3] T. Ya. AZIZOV AND I.S. IOKHVIDOV, Linear Operators in Spaces with an Indefinite Metric, Nauka, Moscow, 1986, English translation: Wiley, New York, 1989.

[4] M. FUJII, T. FURUTAAND E. KAMEI, Furuta’s inequality and its application to Ando’s theorem, Linear Algebra Appl., 179 (1993), 161–169.

[5] T. FURUTA, Applications of order preserving operator inequalities, Op. Theory Adv. Appl., 59 (1992), 180-190.

[6] T. SANO, Furuta inequality of indefinite type, Math. Inequal. Appl., 10 (2007), 381–387.

[7] M. UCHIYAMA, Some exponential operator inequalities, Math. Inequal. Appl., 2 (1999), 469–471.